本系列文章致力于实现“手搓有限元，干翻Ansys的目标”，基本框架为前端显示使用QT实现交互，后端计算采用Visual Studio C++。

目录

Matrix类

1、public function

1.1、构造函数与析构函数

1.2、获取矩阵数值

1.3、设置矩阵

1.4、矩阵转置、单位化

1.5、矩阵的删除与替换

1.6、矩阵初等变换

1.7、矩阵加法

1.8、矩阵乘法

1.9、行列式相关操作

1.10、矩阵求逆

2、private variable

3、全部源码

---

Matrix类

矩阵基本类，用于有限元矩阵计算。

1、public function

公共成员函数，调用可实现基本运算

1.1、构造函数与析构函数

构造函数用来初始化矩阵，析构函数用来释放内存。

Matrix.h声明文件：

	/*
	函数名称:		无参构造函数
	*/
	Matrix();

	/*
	函数名称:		矩阵有参构造函数，初始化为row行、col列的0矩阵
	row:			矩阵行数
	col:			矩阵列数
	*/
	Matrix(int row, int col);

	/*
	函数名称:		矩阵有参构造函数，初始化为row行、col列、数值为mat的矩阵
	row:			矩阵行数
	col:			矩阵列数
	*mat:			矩阵数值一维数组
	*/
	Matrix(int row, int col, double* mat);

	/*
	函数名称:		深拷贝构造函数
	mat:			需要复制的矩阵
	*/
	Matrix(const Matrix& mat);

	/*
	函数名称:		析构函数
	*/
	~Matrix();

Matrix.cpp函数实现文件：

Matrix::Matrix()
{

}

//初始化矩阵 默认值为0
Matrix::Matrix(int row, int col)
{
	this->m_Row = row;
	this->m_Col = col;

	//开辟内存
	this->m_Matrix = new double* [row];
	for (int i = 0; i < row; i++)
	{
		this->m_Matrix[i] = new double[col] {0.0};
	}

}

//初始化矩阵 设定数值
Matrix::Matrix(int row, int col, double *mat)
{
	this->m_Row = row;
	this->m_Col = col;

	//开辟内存
	this->m_Matrix = new double* [row];
	for (int i = 0; i < row; i++)
	{
		this->m_Matrix[i] = new double[col] {0.0};
	}

	//矩阵赋值
	for(int i = 0; i<row; i++)
	{
		for (int j = 0; j < col; j++)
		{
			this->m_Matrix[i][j] = mat[i * col + j];
		}
	}
}

//深拷贝
Matrix::Matrix(const Matrix& mat)
{
	//行列传递
	this->m_Row = mat.m_Row;
	this->m_Col = mat.m_Col;

	//矩阵深拷贝
	this->m_Matrix = new double* [this->m_Row];
	for (int i = 0; i < this->m_Row; i++)
	{
		this->m_Matrix[i] = new double[this->m_Col];
		memcpy(this->m_Matrix[i], mat.m_Matrix[i], sizeof(double) * this->m_Col);
	}
}

//析构函数
Matrix::~Matrix()
{
	//释放矩阵每一行
	for (int i = 0; i < this->m_Row; i++)
	{
		if (this->m_Matrix[i] != NULL)
		{
			delete[]this->m_Matrix[i];
			this->m_Matrix[i] = NULL;
		}
	}

	//释放矩阵顶点
	if (this->m_Matrix != NULL)
	{
		delete[]this->m_Matrix;
		this->m_Matrix = NULL;
	}
}

1.2、获取矩阵数值

可以获取矩阵指定位置数值、打印矩阵。

Matrix.h声明文件：

	//*******************获取矩阵*****************//
	/*
	函数名称:		获取矩阵的第row行、第col列元素数值
	row:			矩阵行数
	col:			矩阵列数
	*/
	double GetMatrixEle(int row, int col);

	/*
	函数名称:		打印矩阵
	*/
	void PrintMat();

Matrix.cpp函数实现文件：

//获取矩阵某个元素 某行某列
double Matrix::GetMatrixEle(int row, int col)
{
	if (row >= this->m_Row)
	{
		std::cout << "Error: <GetMatrixEle> Input row >= m_Row" << std::endl;
		return 0.0;
	}
	else if (col >= this->m_Col)
	{
		std::cout << "Error: <GetMatrixEle> Input col >= m_Col" << std::endl;
		return 0.0;
	}
	else
	{
		return this->m_Matrix[row][col];
	}
}

//矩阵输出
void Matrix::PrintMat()
{
	for (int i = 0; i < this->m_Row; i++)
	{
		for (int j = 0; j < this->m_Col; j++)
		{
			std::cout.setf(std::ios::scientific);		//科学计数法表示
			std::cout << this->m_Matrix[i][j] << "\t";
		}
		std::cout << std::endl;
	}
	std::cout << std::endl;
}

测试验证：

测试代码：

#include "Matrix.h"
int main()
{
	//定义矩阵数值
	double tempValue[9] = {
	1.0, 2.0, 3.0,
	4.0, 5.0, 6.0,
	7.0, 8.0, 0.0
	};

	//创建矩阵
	Matrix* tempMatrix = new Matrix(3, 3, tempValue);

	//打印矩阵
	tempMatrix->PrintMat();

	system("pause");
	return 0;
}

应用输出：

1.000000e+00    2.000000e+00    3.000000e+00
4.000000e+00    5.000000e+00    6.000000e+00
7.000000e+00    8.000000e+00    0.000000e+00

请按任意键继续. . .

1.3、设置矩阵

可进行设置矩阵指定位置数值，以及深拷贝矩阵。

Matrix.h声明文件：

	/*
	函数名称:		设置矩阵第row行、第col列数值
	row:			矩阵行数
	col:			矩阵列数
	value:			设置的矩阵数值
	*/
	void SetMatrixEle(int row, int col, double value);

	/*
	函数名称:		深拷贝矩阵
	mat:			需要复制的矩阵
	*/
	Matrix CopyMat(const Matrix mat);

Matrix.cpp函数实现文件：

//*******************设置矩阵*****************//
void Matrix::SetMatrixEle(int row, int col, double value)
{
	if (row >= this->m_Row)
	{
		std::cout << "Error: <SetMatrixEle> Input row >= m_Row" << std::endl;
		return;
	}
	else if (col >= this->m_Col)
	{
		std::cout << "Error: <SetMatrixEle> Input col >= m_Col" << std::endl;
		return;
	}
	else
	{
		this->m_Matrix[row][col] = value;
		return;
	}
}

//深拷贝矩阵
Matrix Matrix::CopyMat(const Matrix mat)
{
	//行列传递
	this->m_Row = mat.m_Row;
	this->m_Col = mat.m_Col;

	//矩阵深拷贝
	this->m_Matrix = new double* [this->m_Row];
	for (int i = 0; i < this->m_Row; i++)
	{
		this->m_Matrix[i] = new double[this->m_Col];
		memcpy(this->m_Matrix[i], mat.m_Matrix[i], sizeof(double) * this->m_Col);
	}

	return *this;
}

测试验证：

测试代码：

int main()
{
	//定义矩阵数值
	double tempValue[9] = {
	1.0, 2.0, 3.0,
	4.0, 5.0, 6.0,
	7.0, 8.0, 0.0
	};

	//创建矩阵
	Matrix* tempMatrix = new Matrix(3, 3, tempValue);

	//打印矩阵
	std::cout << "数值更改前:" << std::endl;
	tempMatrix->PrintMat();

	//更改特定值
	tempMatrix->SetMatrixEle(1, 1, 10.0);

	//打印矩阵
	std::cout << "数值更改后:" << std::endl;
	tempMatrix->PrintMat();


	system("pause");
	return 0;
}

应用输出：

数值更改前:
1.000000e+00    2.000000e+00    3.000000e+00
4.000000e+00    5.000000e+00    6.000000e+00
7.000000e+00    8.000000e+00    0.000000e+00

数值更改后:
1.000000e+00    2.000000e+00    3.000000e+00
4.000000e+00    1.000000e+01    6.000000e+00
7.000000e+00    8.000000e+00    0.000000e+00

请按任意键继续. . .

1.4、矩阵转置、单位化

可进行矩阵转置，单位化，注意返回值类型为自身的引用，可实现链式编程。

Matrix.h声明文件：

	/*
	函数名称:		矩阵转置，返回的是自身引用，可链式调用
	*/
	Matrix& Transpose();

	/*
	函数名称:		等维度的单位矩阵，前提是方阵
	*/
	Matrix& Uint();

Matrix.cpp函数实现文件：

//矩阵转置
Matrix& Matrix::Transpose()
{
	Matrix* resMat = new Matrix(this->m_Col, this->m_Row);

	for (int i = 0; i < this->m_Row; i++)
	{
		for (int j = 0; j < this->m_Col; j++)
		{
			resMat->m_Matrix[j][i] = this->m_Matrix[i][j];
		}
	}

	return *resMat;
}

//求等长度单位矩阵
Matrix& Matrix::Uint()
{
	//矩阵是否为方阵
	if (this->m_Col != this->m_Row)
	{
		std::cout << "Error: <Uint> Row != Col" << std::endl;

		Matrix* resMat = new Matrix(this->m_Row, this->m_Row);
		return *resMat;
	}
	else
	{
		//单位矩阵初始化
		Matrix* resMat = new Matrix(this->m_Row, this->m_Col);

		//单位矩阵生成
		for (int i = 0; i < this->m_Row; i++)
		{
			resMat->m_Matrix[i][i] = 1.0;
		}

		return *resMat;
	}
}

测试验证：

测试代码：

int main()
{
	//定义矩阵数值
	double tempValue[9] = {
	1.0, 2.0, 3.0,
	4.0, 5.0, 6.0,
	7.0, 8.0, 0.0
	};

	//创建矩阵
	Matrix* tempMatrix = new Matrix(3, 3, tempValue);

	//打印矩阵
	std::cout << "数值转置前:" << std::endl;
	tempMatrix->PrintMat();

	//打印矩阵(注意可链式编程)
	std::cout << "数值转置后:" << std::endl;
	tempMatrix->Transpose().PrintMat();


	system("pause");
	return 0;
}

应用输出：

数值转置前:
1.000000e+00    2.000000e+00    3.000000e+00
4.000000e+00    5.000000e+00    6.000000e+00
7.000000e+00    8.000000e+00    0.000000e+00

数值转置后:
1.000000e+00    4.000000e+00    7.000000e+00
2.000000e+00    5.000000e+00    8.000000e+00
3.000000e+00    6.000000e+00    0.000000e+00

请按任意键继续. . .

1.5、矩阵的删除与替换

可进行矩阵指定行、列的删除与替换，注意返回值类型为自身的引用，可实现链式编程。

Matrix.h声明文件：

	/*
	函数名称:		剔除矩阵中以index为行标和列标的行和列，num代表index的大小
	*index:			矩阵中的行号与列号一维数组
	num:			index动态数组长度
	*/
	Matrix& DeleteMat(int *index, int num);

	/*
	函数名称:		剔除矩阵中以index为行标和列标的行和列，num代表index的大小
	*index:			矩阵中的行号与列号一维动态数组
	num:			index动态数组长度
	*/
	Matrix& DeleteMat(std::vector<int> index, int num);

	/*
	函数名称:		剔除矩阵中以index为行标的行，num代表index的大小
	*index:			矩阵中的行号一维数组
	num:			index动态数组长度
	*/
	Matrix& DeleteRow(int* index, int num);

	/*
	函数名称:		剔除矩阵中以index为行标的行，num代表index的大小
	*index:			矩阵中的行号一维动态数组
	num:			index动态数组长度
	*/
	Matrix& DeleteRow(std::vector<int> index, int num);

	/*
	函数名称:		剔除矩阵中以index为列标的列，num代表index的大小
	*index:			矩阵中的列号一维数组
	num:			index动态数组长度
	*/
	Matrix& DeleteCol(int* index, int num);

	/*
	函数名称:		剔除矩阵中以index为列标的列，num代表index的大小
	*index:			矩阵中的列号一维动态数组
	num:			index动态数组长度
	*/
	Matrix& DeleteCol(std::vector<int> index, int num);

	//******************矩阵的替换****************//
	/*
	函数名称:		替换矩阵中行标和列标为 index中的行与列，num代表index的大小, mat是需要替换的矩阵
	*index:			矩阵中的行标和列标的一维数组
	num:			index动态数组长度
	mat:			需要替换的矩阵
	*/
	Matrix& ReplaceMat(int* index, int num, Matrix& mat);

	/*
	函数名称:		替换矩阵中行标和列标为 index中的行与列，num代表index的大小, mat是需要替换的矩阵
	*index:			矩阵中的行标和列标的一维动态数组
	num:			index动态数组长度
	mat:			需要替换的矩阵
	*/
	Matrix& ReplaceMat(std::vector<int> index, int num, Matrix& mat);

	/*
	函数名称:		替换矩阵中行标为 index中的行，num代表index的大小, mat是需要替换的矩阵
	*index:			矩阵中的行标的一维数组
	num:			index动态数组长度
	mat:			需要替换的矩阵
	*/
	Matrix& ReplaceRow(int* index, int num, Matrix& mat);

	/*
	函数名称:		替换矩阵中行标为 index中的行，num代表index的大小, mat是需要替换的矩阵
	*index:			矩阵中的行标的一动态维数组
	num:			index动态数组长度
	mat:			需要替换的矩阵
	*/
	Matrix& ReplaceRow(std::vector<int> index, int num, Matrix& mat);

	/*
	函数名称:		替换矩阵中列标为 index中的列，num代表index的大小, mat是需要替换的矩阵
	*index:			矩阵中的列标的一维数组
	num:			index动态数组长度
	mat:			需要替换的矩阵
	*/
	Matrix& ReplaceCol(int* index, int num, Matrix& mat);

	/*
	函数名称:		替换矩阵中列标为 index中的列，num代表index的大小, mat是需要替换的矩阵
	*index:			矩阵中的列标的一维动态数组
	num:			index动态数组长度
	mat:			需要替换的矩阵
	*/
	Matrix& ReplaceCol(std::vector<int> index, int num, Matrix& mat);

Matrix.cpp函数实现文件：

//****************矩阵保留与剔除**************//
//剔除矩阵的 index中的行与列，num代表index的大小
Matrix& Matrix::DeleteMat(int* index, int num)
{
	//结果矩阵
	Matrix* resMat = new Matrix(this->m_Row - num, this->m_Col - num);

	int recIndex[MAX_COUNT];
	int currIndex = 0;

	//检验数据有效性
	for (int i = 0; i < num; i++)
	{
		//越界判定
		if (index[i] >= this->m_Row)
		{
			std::cout << "Error: <DeleteMat> Input index[" << i << "] = " << index[i] << " >= m_Row" << std::endl;
			return *this;
		}
		else if (index[i] >= this->m_Col)
		{
			std::cout << "Error: <DeleteMat> Input index[" << i << "] = " << index[i] << " >= m_Col" << std::endl;
			return *this;
		}
	}

	//筛选出剔除后行数
	for (int iRow = 0; iRow < this->m_Row; iRow++)
	{
		for (int iNum = 0; iNum < num; iNum++)
		{
			if (iRow == index[iNum])
			{
				break;
			}

			if (iNum == num-1)
			{
				recIndex[currIndex++] = iRow;
			}
		}
	}

	//加入元素
	for (int iRow = 0; iRow < resMat->m_Row; iRow++)
	{
		for (int iCol = 0; iCol < resMat->m_Col; iCol++)
		{
			resMat->m_Matrix[iRow][iCol] = this->m_Matrix[recIndex[iRow]][recIndex[iCol]];
		}
	}

	return *resMat;
	
}

Matrix& Matrix::DeleteMat(std::vector<int> index, int num)
{
	//结果矩阵
	Matrix* resMat = new Matrix(this->m_Row - num, this->m_Col - num);

	int recIndex[MAX_COUNT];
	int currIndex = 0;

	//检验数据有效性
	for (int i = 0; i < num; i++)
	{
		//越界判定
		if (index[i] >= this->m_Row)
		{
			std::cout << "Error: <DeleteMat> Input index[" << i << "] = " << index[i] << " >= m_Row" << std::endl;
			return *this;
		}
		else if (index[i] >= this->m_Col)
		{
			std::cout << "Error: <DeleteMat> Input index[" << i << "] = " << index[i] << " >= m_Col" << std::endl;
			return *this;
		}
	}

	//筛选出剔除后行数
	for (int iRow = 0; iRow < this->m_Row; iRow++)
	{
		for (int iNum = 0; iNum < num; iNum++)
		{
			if (iRow == index[iNum])
			{
				break;
			}

			if (iNum == num - 1)
			{
				recIndex[currIndex++] = iRow;
			}
		}
	}

	//加入元素
	for (int iRow = 0; iRow < resMat->m_Row; iRow++)
	{
		for (int iCol = 0; iCol < resMat->m_Col; iCol++)
		{
			resMat->m_Matrix[iRow][iCol] = this->m_Matrix[recIndex[iRow]][recIndex[iCol]];
		}
	}

	return *resMat;
}

//剔除矩阵的 index中的行，num代表index的大小
Matrix& Matrix::DeleteRow(int* index, int num)
{
	//结果矩阵
	Matrix* resMat = new Matrix(this->m_Row - num, this->m_Col);

	int recIndex[MAX_COUNT];
	int currIndex = 0;

	//检验数据有效性
	for (int i = 0; i < num; i++)
	{
		//越界判定
		if (index[i] >= this->m_Row)
		{
			std::cout << "Error: <DeleteMat> Input index[" << i << "] = " << index[i] << " >= m_Row" << std::endl;
			return *this;
		}
	}

	//筛选出剔除后行数
	for (int iRow = 0; iRow < this->m_Row; iRow++)
	{
		for (int iNum = 0; iNum < num; iNum++)
		{
			if (iRow == index[iNum])
			{
				break;
			}

			if (iNum == num - 1)
			{
				recIndex[currIndex++] = iRow;
			}
		}
	}

	//加入元素
	for (int iRow = 0; iRow < resMat->m_Row; iRow++)
	{
		for (int iCol = 0; iCol < resMat->m_Col; iCol++)
		{
			resMat->m_Matrix[iRow][iCol] = this->m_Matrix[recIndex[iRow]][iCol];
		}
	}

	return *resMat;
}

Matrix& Matrix::DeleteRow(std::vector<int> index, int num)
{
	//结果矩阵
	Matrix* resMat = new Matrix(this->m_Row - num, this->m_Col);

	int recIndex[MAX_COUNT];
	int currIndex = 0;

	//检验数据有效性
	for (int i = 0; i < num; i++)
	{
		//越界判定
		if (index[i] >= this->m_Row)
		{
			std::cout << "Error: <DeleteMat> Input index[" << i << "] = " << index[i] << " >= m_Row" << std::endl;
			return *this;
		}
	}

	//筛选出剔除后行数
	for (int iRow = 0; iRow < this->m_Row; iRow++)
	{
		for (int iNum = 0; iNum < num; iNum++)
		{
			if (iRow == index[iNum])
			{
				break;
			}

			if (iNum == num - 1)
			{
				recIndex[currIndex++] = iRow;
			}
		}
	}

	//加入元素
	for (int iRow = 0; iRow < resMat->m_Row; iRow++)
	{
		for (int iCol = 0; iCol < resMat->m_Col; iCol++)
		{
			resMat->m_Matrix[iRow][iCol] = this->m_Matrix[recIndex[iRow]][iCol];
		}
	}

	return *resMat;
}

Matrix& Matrix::DeleteCol(int* index, int num)
{
	//结果矩阵
	Matrix* resMat = new Matrix(this->m_Row, this->m_Col - num);

	int recIndex[MAX_COUNT];
	int currIndex = 0;

	//检验数据有效性
	for (int i = 0; i < num; i++)
	{
		//越界判定
		if (index[i] >= this->m_Row)
		{
			std::cout << "Error: <DeleteMat> Input index[" << i << "] = " << index[i] << " >= m_Row" << std::endl;
			return *this;
		}
	}

	//筛选出剔除后行数
	for (int iRow = 0; iRow < this->m_Row; iRow++)
	{
		for (int iNum = 0; iNum < num; iNum++)
		{
			if (iRow == index[iNum])
			{
				break;
			}

			if (iNum == num - 1)
			{
				recIndex[currIndex++] = iRow;
			}
		}
	}

	//加入元素
	for (int iRow = 0; iRow < resMat->m_Row; iRow++)
	{
		for (int iCol = 0; iCol < resMat->m_Col; iCol++)
		{
			resMat->m_Matrix[iRow][iCol] = this->m_Matrix[iRow][recIndex[iCol]];
		}
	}

	return *resMat;
}

Matrix& Matrix::DeleteCol(std::vector<int> index, int num)
{
	//结果矩阵
	Matrix* resMat = new Matrix(this->m_Row, this->m_Col - num);

	int recIndex[MAX_COUNT];
	int currIndex = 0;

	//检验数据有效性
	for (int i = 0; i < num; i++)
	{
		//越界判定
		if (index[i] >= this->m_Row)
		{
			std::cout << "Error: <DeleteMat> Input index[" << i << "] = " << index[i] << " >= m_Row" << std::endl;
			return *this;
		}
	}

	//筛选出剔除后行数
	for (int iRow = 0; iRow < this->m_Row; iRow++)
	{
		for (int iNum = 0; iNum < num; iNum++)
		{
			if (iRow == index[iNum])
			{
				break;
			}

			if (iNum == num - 1)
			{
				recIndex[currIndex++] = iRow;
			}
		}
	}

	//加入元素
	for (int iRow = 0; iRow < resMat->m_Row; iRow++)
	{
		for (int iCol = 0; iCol < resMat->m_Col; iCol++)
		{
			resMat->m_Matrix[iRow][iCol] = this->m_Matrix[iRow][recIndex[iCol]];
		}
	}

	return *resMat;
}

//******************矩阵的替换****************//
//替换矩阵中的行和列 index中的行与列，num代表index的大小
Matrix& Matrix::ReplaceMat(int* index, int num, Matrix& mat)
{

	//错误判定 方阵
	if (this->m_Row != this->m_Col)
	{
		std::cout << "Error: <ReplaceMat> this m_Col != m_Row" << std::endl;
		return *this;
	}

	//检验插入矩阵为方阵
	if (mat.m_Row != mat.m_Col)
	{
		std::cout << "Error: <ReplaceMat> mat m_Col != m_Row" << std::endl;
		return *this;
	}

	//检验插入矩阵大小与num保持一致
	if (mat.m_Col != num)
	{
		std::cout << "Error: <ReplaceMat> num != mat.m_Col" << std::endl;
		return *this;
	}

	//检验数据有效性
	for (int i = 0; i < num; i++)
	{
		//越界判定
		if (index[i] >= this->m_Row)
		{
			std::cout << "Error: <ReplaceMat> Input index[" << i << "] = " << index[i] << " >= m_Row" << std::endl;
			return *this;
		}
		else if (index[i] >= this->m_Col)
		{
			std::cout << "Error: <ReplaceMat> Input index[" << i << "] = " << index[i] << " >= m_Col" << std::endl;
			return *this;
		}
	}

	//结果矩阵
	Matrix* resMat = new Matrix(*this);

	//加入元素
	for (int iRow = 0; iRow < num; iRow++)
	{
		for (int iCol = 0; iCol < num; iCol++)
		{
			resMat->m_Matrix[index[iRow]][index[iCol]] = mat.m_Matrix[iRow][iCol];
		}
	}

	return *resMat;
}

Matrix& Matrix::ReplaceMat(std::vector<int> index, int num, Matrix& mat)
{
	//错误判定 方阵
	if (this->m_Row != this->m_Col)
	{
		std::cout << "Error: <ReplaceMat> this m_Col != m_Row" << std::endl;
		return *this;
	}

	//检验插入矩阵为方阵
	if (mat.m_Row != mat.m_Col)
	{
		std::cout << "Error: <ReplaceMat> mat m_Col != m_Row" << std::endl;
		return *this;
	}

	//检验插入矩阵大小与num保持一致
	if (mat.m_Col != num)
	{
		std::cout << "Error: <ReplaceMat> num != mat.m_Col" << std::endl;
		return *this;
	}

	//检验数据有效性
	for (int i = 0; i < num; i++)
	{
		//越界判定
		if (index[i] >= this->m_Row)
		{
			std::cout << "Error: <ReplaceMat> Input index[" << i << "] = " << index[i] << " >= m_Row" << std::endl;
			return *this;
		}
		else if (index[i] >= this->m_Col)
		{
			std::cout << "Error: <ReplaceMat> Input index[" << i << "] = " << index[i] << " >= m_Col" << std::endl;
			return *this;
		}
	}

	//结果矩阵
	Matrix* resMat = new Matrix(*this);

	//加入元素
	for (int iRow = 0; iRow < num; iRow++)
	{
		for (int iCol = 0; iCol < num; iCol++)
		{
			resMat->m_Matrix[index[iRow]][index[iCol]] = mat.m_Matrix[iRow][iCol];
		}
	}

	return *resMat;
}

//替换矩阵中的行 index中的行，num代表index的大小, mat是需要替换的矩阵
Matrix& Matrix::ReplaceRow(int* index, int num, Matrix& mat)
{
	//检验插入矩阵大小与num保持一致
	if (mat.m_Row != num)
	{
		std::cout << "Error: <ReplaceRow> num != mat.m_Row" << std::endl;
		return *this;
	}

	//检验数据有效性
	for (int i = 0; i < num; i++)
	{
		//越界判定
		if (index[i] >= this->m_Row)
		{
			std::cout << "Error: <ReplaceRow> Input index[" << i << "] = " << index[i] << " >= m_Row" << std::endl;
			return *this;
		}
	}

	//当前矩阵列数应与mat列数一致
	if (this->m_Col != mat.m_Col)
	{
		std::cout << "Error: <ReplaceRow> this->m_Col != mat.m_Col" << std::endl;
		return *this;
	}

	//结果矩阵
	Matrix* resMat = new Matrix(*this);

	//加入元素
	for (int iRow = 0; iRow < num; iRow++)
	{
		for (int iCol = 0; iCol < resMat->m_Col; iCol++)
		{
			resMat->m_Matrix[index[iRow]][iCol] = mat.m_Matrix[iRow][iCol];
		}
	}

	return *resMat;
}

Matrix& Matrix::ReplaceRow(std::vector<int> index, int num, Matrix& mat)
{
	//检验插入矩阵大小与num保持一致
	if (mat.m_Row != num)
	{
		std::cout << "Error: <ReplaceRow> num != mat.m_Row" << std::endl;
		return *this;
	}

	//检验数据有效性
	for (int i = 0; i < num; i++)
	{
		//越界判定
		if (index[i] >= this->m_Row)
		{
			std::cout << "Error: <ReplaceRow> Input index[" << i << "] = " << index[i] << " >= m_Row" << std::endl;
			return *this;
		}
	}

	//当前矩阵列数应与mat列数一致
	if (this->m_Col != mat.m_Col)
	{
		std::cout << "Error: <ReplaceRow> this->m_Col != mat.m_Col" << std::endl;
		return *this;
	}

	//结果矩阵
	Matrix* resMat = new Matrix(*this);

	//加入元素
	for (int iRow = 0; iRow < num; iRow++)
	{
		for (int iCol = 0; iCol < resMat->m_Col; iCol++)
		{
			resMat->m_Matrix[index[iRow]][iCol] = mat.m_Matrix[iRow][iCol];
		}
	}

	return *resMat;
}

//替换矩阵中的列 index中的列，num代表index的大小, mat是需要替换的矩阵
Matrix& Matrix::ReplaceCol(int* index, int num, Matrix& mat)
{
	//检验插入矩阵大小与num保持一致
	if (mat.m_Col != num)
	{
		std::cout << "Error: <ReplaceCol> mat.m_Col != num" << std::endl;
		return *this;
	}

	//检验数据有效性
	for (int i = 0; i < num; i++)
	{
		//越界判定
		if (index[i] >= this->m_Col)
		{
			std::cout << "Error: <ReplaceCol> Input index[" << i << "] = " << index[i] << " >= m_Col" << std::endl;
			return *this;
		}
	}

	//当前矩阵行数应与mat行数一致
	if (this->m_Row != mat.m_Row)
	{
		std::cout << "Error: <ReplaceCol> this->m_Row != mat.m_Row" << std::endl;
		return *this;
	}

	//结果矩阵
	Matrix* resMat = new Matrix(*this);

	//加入元素
	for (int iRow = 0; iRow < resMat->m_Row; iRow++)
	{
		for (int iCol = 0; iCol < num; iCol++)
		{
			resMat->m_Matrix[iRow][index[iCol]] = mat.m_Matrix[iRow][iCol];
		}
	}

	return *resMat;
}

Matrix& Matrix::ReplaceCol(std::vector<int> index, int num, Matrix& mat)
{
	//检验插入矩阵大小与num保持一致
	if (mat.m_Col != num)
	{
		std::cout << "Error: <ReplaceCol> mat.m_Col != num" << std::endl;
		return *this;
	}

	//检验数据有效性
	for (int i = 0; i < num; i++)
	{
		//越界判定
		if (index[i] >= this->m_Col)
		{
			std::cout << "Error: <ReplaceCol> Input index[" << i << "] = " << index[i] << " >= m_Col" << std::endl;
			return *this;
		}
	}

	//当前矩阵行数应与mat行数一致
	if (this->m_Row != mat.m_Row)
	{
		std::cout << "Error: <ReplaceCol> this->m_Row != mat.m_Row" << std::endl;
		return *this;
	}

	//结果矩阵
	Matrix* resMat = new Matrix(*this);

	//加入元素
	for (int iRow = 0; iRow < resMat->m_Row; iRow++)
	{
		for (int iCol = 0; iCol < num; iCol++)
		{
			resMat->m_Matrix[iRow][index[iCol]] = mat.m_Matrix[iRow][iCol];
		}
	}

	return *resMat;
}

测试验证：

测试代码：

int main()
{
	//定义矩阵数值
	double tempValue[9] = {
	1.0, 2.0, 3.0,
	4.0, 5.0, 6.0,
	7.0, 8.0, 0.0
	};

	//替换数值
	double replaceValue[3] = {
	1.42, 2.54, 9.65
	};

	//创建矩阵
	Matrix* tempMatrix = new Matrix(3, 3, tempValue);
	Matrix* tempReplaceMatrix = new Matrix(1, 3, replaceValue);

	int replaceCol[1] = {
	2
	};

	//打印矩阵
	std::cout << "数值第3行替换前:" << std::endl;
	tempMatrix->PrintMat();

	//打印矩阵(注意可链式编程)
	std::cout << "数值第3行替换后:" << std::endl;
	tempMatrix->ReplaceRow(replaceCol, 1, *tempReplaceMatrix).PrintMat();

	//打印矩阵
	std::cout << "数值第3行删除前:" << std::endl;
	tempMatrix->PrintMat();

	//打印矩阵(注意可链式编程)
	std::cout << "数值第3行删除后:" << std::endl;
	tempMatrix->DeleteRow(replaceCol, 1).PrintMat();


	system("pause");
	return 0;
}

应用输出：

数值第3行替换前:
1.000000e+00    2.000000e+00    3.000000e+00
4.000000e+00    5.000000e+00    6.000000e+00
7.000000e+00    8.000000e+00    0.000000e+00

数值第3行替换后:
1.000000e+00    2.000000e+00    3.000000e+00
4.000000e+00    5.000000e+00    6.000000e+00
1.420000e+00    2.540000e+00    9.650000e+00

数值第3行删除前:
1.000000e+00    2.000000e+00    3.000000e+00
4.000000e+00    5.000000e+00    6.000000e+00
7.000000e+00    8.000000e+00    0.000000e+00

数值第3行删除后:
1.000000e+00    2.000000e+00    3.000000e+00
4.000000e+00    5.000000e+00    6.000000e+00

请按任意键继续. . .

1.6、矩阵初等变换

可实现矩阵的初等变化，注意返回值类型为自身的引用，可实现链式编程。

Matrix.h声明文件：

	//*****************矩阵初等变化***************//
	/*
	函数名称:		交换矩阵中行标为row0与row1的元素
	row0:			矩阵行标0
	row1:			矩阵行标1
	*/
	Matrix& SwapRow(int row0, int row1);

	/*
	函数名称:		交换矩阵中列标为col0与col1的元素
	col0:			矩阵列标0
	col1:			矩阵列标1
	*/
	Matrix& SwapCol(int col0, int col1);

	/*
	函数名称:		矩阵行加法 rowLocal = rowLocal + rate *rowAdd
	rowLocal:		矩阵行标，被加数
	rowAdd:			矩阵行标，加数
	rate:			加数前倍数
	*/
	Matrix& AddRow(int rowLocal, int rowAdd, double rate = 1.0);

	//矩阵加法 某列 + 倍数*某列
	/*
	函数名称:		矩阵列加法 colLocal = colLocal + rate * colAdd
	colLocal:		矩阵列标，被加数
	colAdd:			矩阵列标，加数
	rate:			加数前倍数
	*/
	Matrix& AddCol(int colLocal, int colAdd, double rate = 1.0);

	//*******************矩阵加法*****************//
	/*
	函数名称:		矩阵加法 本矩阵 = 本矩阵 + mat 前提是两个矩阵维度一致
	mat:			加数矩阵
	*/
	Matrix& AddMat(Matrix& mat);

Matrix.cpp函数实现文件：

//*****************矩阵初等变化***************//
Matrix& Matrix::SwapRow(int row0, int row1)
{
	//错误判定 越界
	if ((this->m_Row <= row0) || (this->m_Col <= row1))
	{
		std::cout << "Error: <SwapRow> Input row0 Or row1 More Than m_Row" << std::endl;
		return *this;
	}
	else if ((0 > row0) || (0 > row1))
	{
		std::cout << "Error: <SwapRow> Input row0 Or row1 Less 0" << std::endl;
		return *this;
	}
	else
	{
		//结果矩阵初始化
		Matrix* resMat = new Matrix(*this);

		//中转临时变量
		double temp = 0.0;

		for (int j = 0; j < resMat->m_Col; j++)
		{
			temp = resMat->m_Matrix[row0][j];
			resMat->m_Matrix[row0][j] = resMat->m_Matrix[row1][j];
			resMat->m_Matrix[row1][j] = temp;
		}

		return*resMat;
	}
}

Matrix& Matrix::SwapCol(int col0, int col1)
{
	//错误判定 越界
	if ((this->m_Col <= col0) || (this->m_Col <= col1))
	{
		std::cout << "Error: <SwapCol> Input col0 Or col1 More Than m_Col" << std::endl;
		return *this;
	}
	else if ((0 > col0) || (0 > col1))
	{
		std::cout << "Error: <SwapCol> Input col0 Or col1 Less 0" << std::endl;
		return *this;
	}
	else
	{
		//结果矩阵初始化
		Matrix* resMat = new Matrix(*this);

		//中转临时变量
		double temp = 0.0;

		for (int i = 0; i < resMat->m_Row; i++)
		{
			temp = resMat->m_Matrix[i][col0];
			resMat->m_Matrix[i][col0] = resMat->m_Matrix[i][col1];
			resMat->m_Matrix[i][col1] = temp;
		}

		return*resMat;
	}
}

//矩阵加法 某行 + 倍数*某行
Matrix& Matrix::AddRow(int rowLocal, int rowAdd, double rate)
{
	if ((this->m_Row <= rowLocal) || (this->m_Row <= rowAdd))
	{
		std::cout << "Error: <AddRow> Input rowLocal Or rowAdd More Than m_Row" << std::endl;
		return *this;
	}
	else if ((0 > rowLocal) || (0 > rowAdd))
	{
		std::cout << "Error: <AddRow> Input rowLocal Or rowAdd Less 0" << std::endl;
		return *this;
	}
	else
	{
		//结果矩阵初始化
		Matrix* resMat = new Matrix(*this);

		//指定行相加
		for (int j = 0; j < resMat->m_Col; j++)
		{
			resMat->m_Matrix[rowLocal][j] += rate * resMat->m_Matrix[rowAdd][j];
		}

		return *resMat;
	}
}

//矩阵加法 某列 + 倍数*某列
Matrix& Matrix::AddCol(int colLocal, int colAdd, double rate)
{
	if ((this->m_Col <= colLocal) || (this->m_Col <= colAdd))
	{
		std::cout << "Error: <AddCol> Input colLocal Or colAdd More Than m_Col" << std::endl;
		return *this;
	}
	else if ((0 > colLocal) || (0 > colAdd))
	{
		std::cout << "Error: <AddCol> Input colLocal Or colAdd Less 0" << std::endl;
		return *this;
	}
	else
	{
		//结果矩阵初始化
		Matrix* resMat = new Matrix(*this);

		//指定列相加
		for (int i = 0; i < resMat->m_Row; i++)
		{
			resMat->m_Matrix[i][colLocal] += rate * resMat->m_Matrix[i][colAdd];
		}

		return *resMat;
	}
}

测试验证：

测试代码：

int main()
{
	//定义矩阵数值
	double tempValue[9] = {
	1.0, 2.0, 3.0,
	4.0, 5.0, 6.0,
	7.0, 8.0, 0.0
	};

	//创建矩阵
	Matrix* tempMatrix = new Matrix(3, 3, tempValue);

	//打印矩阵
	std::cout << "************************" << std::endl;
	std::cout << "数值第1行与第3行交换前:" << std::endl;
	tempMatrix->PrintMat();

	//打印矩阵(注意可链式编程)
	std::cout << "数值第1行与第3行交换后:" << std::endl;
	tempMatrix->SwapRow(0, 2).PrintMat();

	//打印矩阵
	std::cout << "************************" << std::endl;
	std::cout << "数值第1行与第3行相加前:" << std::endl;
	tempMatrix->PrintMat();

	//打印矩阵(注意可链式编程)
	std::cout << "数值第1行与第3行相加后:" << std::endl;
	tempMatrix->AddRow(0, 2).PrintMat();


	system("pause");
	return 0;
}

应用输出：

************************
数值第1行与第3行交换前:
1.000000e+00    2.000000e+00    3.000000e+00
4.000000e+00    5.000000e+00    6.000000e+00
7.000000e+00    8.000000e+00    0.000000e+00

数值第1行与第3行交换后:
7.000000e+00    8.000000e+00    0.000000e+00
4.000000e+00    5.000000e+00    6.000000e+00
1.000000e+00    2.000000e+00    3.000000e+00

************************
数值第1行与第3行相加前:
1.000000e+00    2.000000e+00    3.000000e+00
4.000000e+00    5.000000e+00    6.000000e+00
7.000000e+00    8.000000e+00    0.000000e+00

数值第1行与第3行相加后:
8.000000e+00    1.000000e+01    3.000000e+00
4.000000e+00    5.000000e+00    6.000000e+00
7.000000e+00    8.000000e+00    0.000000e+00

请按任意键继续. . .

1.7、矩阵加法

实现矩阵基本加法，注意返回值类型为自身的引用，可实现链式编程。

Matrix.h声明文件：

	//*******************矩阵加法*****************//
	/*
	函数名称:		矩阵加法 本矩阵 = 本矩阵 + mat 前提是两个矩阵维度一致
	mat:			加数矩阵
	*/
	Matrix& AddMat(Matrix& mat);

Matrix.cpp函数实现文件：

//*******************矩阵加法*****************//
Matrix& Matrix::AddMat(Matrix& mat)
{
	Matrix* ResMat = new Matrix(*this);

	for (int i = 0; i < ResMat->m_Row; i++)
	{
		for (int j = 0; j < ResMat->m_Col; j++)
		{
			ResMat->m_Matrix[i][j] += mat.m_Matrix[i][j];
		}
	}

	return *ResMat;
}

测试验证：

测试代码：

int main()
{
	//定义矩阵数值
	double tempValue0[9] = {
	1.0, 2.0, 3.0,
	4.0, 5.0, 6.0,
	7.0, 8.0, 0.0
	};

	//定义矩阵数值
	double tempValue1[9] = {
	2.0, 5.0, 8.0,
	1.0, 5.0, 9.0,
	3.0, 6.0, 7.0
	};

	//创建矩阵
	Matrix* tempMatrix0 = new Matrix(3, 3, tempValue0);
	Matrix* tempMatrix1 = new Matrix(3, 3, tempValue1);

	//打印矩阵
	std::cout << "************************" << std::endl;
	std::cout << "数值矩阵相加前:" << std::endl;
	tempMatrix0->PrintMat();

	//打印矩阵(注意可链式编程)
	std::cout << "数值矩阵相加后:" << std::endl;
	tempMatrix0->AddMat(*tempMatrix1).PrintMat();


	system("pause");
	return 0;
}

应用输出：

************************
数值矩阵相加前:
1.000000e+00    2.000000e+00    3.000000e+00
4.000000e+00    5.000000e+00    6.000000e+00
7.000000e+00    8.000000e+00    0.000000e+00

数值矩阵相加后:
3.000000e+00    7.000000e+00    1.100000e+01
5.000000e+00    1.000000e+01    1.500000e+01
1.000000e+01    1.400000e+01    7.000000e+00

请按任意键继续. . .

1.8、矩阵乘法

实现矩阵基本乘法，注意返回值类型为自身的引用，可实现链式编程。

Matrix.h声明文件：

	//*******************矩阵乘法*****************//
	/*
	函数名称:		矩阵乘法 本矩阵 = 本矩阵*num 
	num:			矩阵乘数
	*/
	Matrix& MultNum(double num);

	/*
	函数名称:		矩阵乘法（运算符重载） 本矩阵 = 本矩阵*num 
	num:			矩阵乘数
	*/
	Matrix& operator * (double num);

	/*
	函数名称:		矩阵某行乘数值row = row*num
	num:			矩阵某列乘数
	row:			矩阵行标
	*/
	Matrix& MultRow(double num, int row);

	/*
	函数名称:		矩阵某列乘数值col = col *num
	num:			矩阵某列乘数
	col:			矩阵列标
	*/
	Matrix& MultCol(double num, int col);
	
	/*
	函数名称:		矩阵乘法，按照矩阵相乘规则
	inputMat:		乘数矩阵
	*/
	Matrix& MultMat(Matrix& inputMat);

Matrix.cpp函数实现文件：

//*******************矩阵乘法*****************//
//矩阵数乘
Matrix& Matrix::MultNum(double num)
{
	//结果矩阵初始化
	Matrix* resMat = new Matrix(this->m_Row, this->m_Col);

	//乘后矩阵生成
	for (int i = 0; i < this->m_Row; i++)
	{
		for (int j = 0; j < this->m_Col; j++)
		{
			resMat->m_Matrix[i][j] = num * this->m_Matrix[i][j];
		}
	}

	return *resMat;
}

//运算符重载 矩阵数乘
Matrix& Matrix::operator*(double num)
{
	//结果矩阵初始化
	Matrix* resMat = new Matrix(this->m_Row, this->m_Col);

	//乘后矩阵生成
	for (int i = 0; i < this->m_Row; i++)
	{
		for (int j = 0; j < this->m_Col; j++)
		{
			resMat->m_Matrix[i][j] = num * this->m_Matrix[i][j];
		}
	}

	return *resMat;
}

//矩阵某行乘数值 行标从0开始计数
Matrix& Matrix::MultRow(double num, int row)
{
	if (this->m_Row <= row)
	{
		std::cout << "Error: <MultRow> Input row More Than m_Row" << std::endl;
		return *this;
	}
	else if (0 > row)
	{
		std::cout << "Error: <MultRow> Input row Less 0" << std::endl;
		return *this;
	}
	else
	{
		//结果矩阵初始化
		Matrix* resMat = new Matrix(*this);

		//乘后矩阵生成
		for (int j = 0; j < this->m_Col; j++)
		{
			resMat->m_Matrix[row][j] = num * this->m_Matrix[row][j];
		}

		return *resMat;
	}

}

//矩阵某列乘数值 列标从0开始计数
Matrix& Matrix::MultCol(double num, int col)
{
	if (this->m_Col <= col)
	{
		std::cout << "Error: <MultCol> Input col More Than m_Row" << std::endl;
		return *this;
	}
	else if (0 > col)
	{
		std::cout << "Error: <MultCol> Input col Less 0" << std::endl;
		return *this;
	}
	else
	{
		//结果矩阵初始化
		Matrix* resMat = new Matrix(*this);

		//乘后矩阵生成
		for (int i = 0; i < this->m_Row; i++)
		{
			resMat->m_Matrix[i][col] = num * this->m_Matrix[i][col];
		}

		return *resMat;
	}
}



//矩阵相乘
Matrix& Matrix::MultMat(Matrix& inputMat)
{
	Matrix *resMat = new Matrix(this->m_Row, inputMat.m_Col);

	if (this->m_Col != inputMat.m_Row)
	{
		std::cout << "Matrix Mult Error!" << std::endl;
		return *resMat;
	}
	else
	{
		for (int i = 0; i < this->m_Row; i++)
		{
			for (int j = 0; j < inputMat.m_Col; j++)
			{
				for (int k = 0; k < this->m_Col; k++)
				{
					resMat->m_Matrix[i][j] += this->m_Matrix[i][k] * inputMat.m_Matrix[k][j];
				}
			}
		}

		return *resMat;
	}
}

测试验证：

测试代码：

int main()
{
	//定义矩阵数值
	double tempValue0[9] = {
	1.0, 2.0, 3.0,
	4.0, 5.0, 6.0,
	7.0, 8.0, 0.0
	};

	//定义矩阵数值
	double tempValue1[9] = {
	2.0, 5.0, 8.0,
	1.0, 5.0, 9.0,
	3.0, 6.0, 7.0
	};

	//创建矩阵
	Matrix* tempMatrix0 = new Matrix(3, 3, tempValue0);
	Matrix* tempMatrix1 = new Matrix(3, 3, tempValue1);

	//打印矩阵
	std::cout << "************************" << std::endl;
	std::cout << "数值矩阵相乘前:" << std::endl;
	tempMatrix0->PrintMat();

	//打印矩阵(注意可链式编程)
	std::cout << "数值矩阵相乘后:" << std::endl;
	tempMatrix0->MultMat(*tempMatrix1).PrintMat();


	system("pause");
	return 0;
}

应用输出：

************************
数值矩阵相乘前:
1.000000e+00    2.000000e+00    3.000000e+00
4.000000e+00    5.000000e+00    6.000000e+00
7.000000e+00    8.000000e+00    0.000000e+00

数值矩阵相乘后:
1.300000e+01    3.300000e+01    4.700000e+01
3.100000e+01    8.100000e+01    1.190000e+02
2.200000e+01    7.500000e+01    1.280000e+02

请按任意键继续. . .

matlab验证：

>> tempMatrix0 = [1 2 3;4 5 6; 7 8 0];
>> tempMatrix1 = [2 5 8;1 5 9; 3 6 7];
>> res = tempMatrix0*tempMatrix1

res =

    13    33    47
    31    81   119
    22    75   128

1.9、行列式相关操作

实现行列式计算相关操作。

Matrix.h声明文件：

	//******************行列式相关操作***********************//
	/*
	函数名称:		求解矩阵对应行列式数值，前提为方阵，按照定义求解，时间复杂度为O(n!*n)，一般不用此方法求解
	*/
	double Det();

	/*
	函数名称:		求解矩阵对应行列式的顺序主子式，前提为方阵，按照定义求解，时间复杂度为O(n!*n)，一般不用此方法求解
	order:			阶数
	*/
	double Det(int order);
	 
	/*
	函数名称:		矩阵行标为row、列标为col的余子式
	row:			矩阵行标
	col:			矩阵列标
	*/
	Matrix& ChildMatrix(int row, int col);

	/*
	函数名称:		通过高斯列主消元求解矩阵行列式数值，最为常用
	*/
	double DetRow();

Matrix.cpp函数实现文件：

//矩阵的行列式数值
double Matrix::Det()
{
	double res = 0.0;
	int sign = 1;

	if (this->m_Row != this->m_Col)
	{
		//错误判定
		std::cout << "Error: <Det> Matrix Col != Row" << std::endl;
		return 0;
	}
	else if (this->m_Row <= 1)
	{
		//程序终止出口
		return this->m_Matrix[0][0];
	}
	else
	{
		for (int i = 0; i < this->m_Col; i++)
		{
			Matrix* temp = &(this->ChildMatrix(0, i));
			res += sign * this->m_Matrix[0][i] * (temp->Det());
			sign = -1*sign;
			delete temp;
		}
	}

}

//矩阵行列式顺序主子式 order阶数
double Matrix::Det(int order)
{
	if (this->m_Row != this->m_Col)
	{
		//错误判定
		std::cout << "Error: <Det> Matrix Col != Row" << std::endl;
		return 0;
	}
	else if (order < 0)
	{
		std::cout << "Error: <Det>  Input Order Less 0" << std::endl;
		return 0;
	}
	else if (order >= this->m_Row)
	{
		std::cout << "Error: <Det> Input Order More Than Row" << std::endl;
		return 0;
	}
	else
	{
		Matrix tempMat(order + 1, order + 1);
		for (int i = 0; i < tempMat.m_Col; i++)
		{
			for (int j = 0; j < tempMat.m_Row; j++)
			{
				tempMat.m_Matrix[i][j] = this->m_Matrix[i][j];
			}
		}
		return tempMat.Det();
	}
}

//求解余子式
Matrix& Matrix::ChildMatrix(int row, int col)
{
	if (this->m_Row != this->m_Col)
	{
		std::cout << "Error: <ChildMatrix> Matrix row != col" << std::endl;
		return *this;
	}
	else if (this->m_Row <= 1)
	{
		std::cout << "Error: <ChildMatrix> Matrix Row Less 1 " << std::endl;
		return *this;
	}
	else if ((row > this->m_Row) || (col > this->m_Col))
	{
		std::cout << "Error: <ChildMatrix> Input Row Or Col More Than Matix Max Row Or Col" << std::endl; 
		return* this; 
	}
	else
	{
		Matrix* resMat = new Matrix(this->m_Row-1, this->m_Col-1);

		for (int i = 0; i < this->m_Row; i++)
		{
			for (int j = 0; j < this->m_Col; j++)
			{
				if ((i < row) && (j < col))
					resMat->m_Matrix[i][j] = this->m_Matrix[i][j];
				else if((i > row) && (j < col))
					resMat->m_Matrix[i-1][j] = this->m_Matrix[i][j];
				else if((i < row) && (j > col))
					resMat->m_Matrix[i][j - 1] = this->m_Matrix[i][j];
				else if((i > row) && (j > col))
					resMat->m_Matrix[i - 1][j - 1] = this->m_Matrix[i][j];
			}
		}

		return *resMat;
	}
}

//列主消元处理为上三角矩阵
double Matrix::DetRow()
{
	//交换标志位 1代表偶数次交换 -1代表奇数次交换
	int flagShift = 1;

	//本矩阵
	Matrix *localMat = new Matrix(*this);

	//行列式数值
	double resDet = 1.0;

	//*******************通过交换 num1*i + num2*j 实现下三角为0***************//
	for (int i = 0; i < localMat->m_Row - 1; i++)
	{
		//记录最大行所在行标
		int tempMaxRow = i;

		for (int i1 = i + 1; i1 < localMat->m_Row; i1++)
		{
			if (abs(localMat->m_Matrix[i1][i]) > abs(localMat->m_Matrix[tempMaxRow][i]))
			{
				tempMaxRow = i1;
			}
		}

		if (tempMaxRow != i)
		{
			//std::cout << i << " 行交换" << tempMaxRow << " 行" << std::endl;
			//进行交换 将当前第i行与第tempMaxRow行进行互换 初等行变换
	
			*localMat = localMat->SwapRow(i, tempMaxRow);

			//记录交换次数
			flagShift = -flagShift;

			//localMat->PrintMat();
		}


		//此对角线以下的元素通过初等变化为0
		for (int i2 = i + 1; i2 < localMat->m_Row; i2++)
		{
			if (localMat->m_Matrix[i2][i] != 0)
			{
				//std::cout << "<" << localMat->m_Matrix[i][i] << "> *" << i2 << " 行 + <" << -1.0 * (localMat->m_Matrix[i2][i]) << "> *" << i << " 行" << std::endl;

				*localMat = localMat->AddRow(i2, i, -1.0 * (localMat->m_Matrix[i2][i]) / localMat->m_Matrix[i][i]);

				//localMat->PrintMat();

			}
		}
	}

	//计算行列式数值 对角线相乘
	for (int i = 0; i < localMat->m_Row; i++)
	{
		resDet = resDet * localMat->m_Matrix[i][i];
	}

	//矩阵交换一次就会变号
	resDet = flagShift * resDet;

	//清理localMatrix
	delete localMat;

	return resDet;
}

测试验证：

测试代码：

int main()
{
	//定义矩阵数值
	double tempValue0[9] = {
	1.0, 2.0, 3.0,
	4.0, 5.0, 6.0,
	7.0, 8.0, 0.0
	};


	//创建矩阵
	Matrix* tempMatrix0 = new Matrix(3, 3, tempValue0);

	//打印矩阵
	std::cout << "************************" << std::endl;
	std::cout << "高斯列主消元过程:" << std::endl;
	std::cout << tempMatrix0->DetRow() << std::endl;

	system("pause");
	return 0;
}

应用输出：

************************
高斯列主消元过程:
0 行交换2 行
7.000000e+00    8.000000e+00    0.000000e+00
4.000000e+00    5.000000e+00    6.000000e+00
1.000000e+00    2.000000e+00    3.000000e+00

<7.000000e+00> *1 行 + <-4.000000e+00> *0 行
7.000000e+00    8.000000e+00    0.000000e+00
0.000000e+00    4.285714e-01    6.000000e+00
1.000000e+00    2.000000e+00    3.000000e+00

<7.000000e+00> *2 行 + <-1.000000e+00> *0 行
7.000000e+00    8.000000e+00    0.000000e+00
0.000000e+00    4.285714e-01    6.000000e+00
0.000000e+00    8.571429e-01    3.000000e+00

1 行交换2 行
7.000000e+00    8.000000e+00    0.000000e+00
0.000000e+00    8.571429e-01    3.000000e+00
0.000000e+00    4.285714e-01    6.000000e+00

<8.571429e-01> *2 行 + <-4.285714e-01> *1 行
7.000000e+00    8.000000e+00    0.000000e+00
0.000000e+00    8.571429e-01    3.000000e+00
0.000000e+00    5.551115e-17    4.500000e+00

2.700000e+01
请按任意键继续. . .

Matlab验证：

>> tempMatrix0 = [1 2 3;4 5 6; 7 8 0];
>> det(tempMatrix0)

ans =

   27.0000

1.10、矩阵求逆

实现矩阵求逆相关操作

Matrix.h声明文件：

	//*********************矩阵求逆********************//
	/*
	函数名称:		矩阵求逆，按照定义求解，1/|A|*(A*)，时间复杂度为O(n!*n)，一般不用此方法
	*/
	Matrix& Inverse();

	/*
	函数名称:		矩阵求逆，通过行初等变化，高斯列主消元法求解
	*/
	Matrix& InverseRow();

	/*
	函数名称:		矩阵求逆，只针对于下三角矩阵进行求解
	*/
	Matrix& InverseDownTriangle();

	/*
	函数名称:		矩阵求逆，只针对于上三角矩阵进行求解
	*/
	Matrix& InverseUpTriangle();

	//矩阵LU分解
	/*
	函数名称:		矩阵LU分解
	LMat:			矩阵分解后的L矩阵
	UMat:			矩阵分解后的U矩阵
	*/
	void ResolveLU(Matrix& LMat, Matrix& UMat);

	/*
	函数名称:		矩阵的LUP分解 P*A = L*U 添加了列主消元功能
	LMat:			矩阵分解后的L矩阵
	UMat:			矩阵分解后的U矩阵
	PMat:			矩阵分解后的P矩阵
	*/
	void ResolveLUP(Matrix& LMat, Matrix& UMat, Matrix& PMat);

Matrix.cpp函数实现文件：

//矩阵求逆
Matrix& Matrix::Inverse()
{
	if (abs(this->DetRow()) < MIN_DET)
	{
		std::cout << "Error: <Inverse> Matrix Det Near 0" << std::endl;
		return *this;
	}
	else
	{
		Matrix* resMat = new Matrix(this->m_Row, this->m_Col);
		for (int i = 0; i < this->m_Row; i++)
		{
			for (int j = 0; j < this->m_Col; j++)
			{
				Matrix* temp = &(this->ChildMatrix(j, i));
				resMat->m_Matrix[i][j] = pow(-1.0, (i + j)) / this->DetRow() * (temp->DetRow());
				delete temp;
			}
		}

		return *resMat;
	}
}

//矩阵求逆 行初等变化
Matrix& Matrix::InverseRow()
{
	//错误判断
	if (abs(this->DetRow()) < MIN_DET)
	{
		std::cout << "Error: <InverseRow> Matrix Det Near 0" << std::endl;
		return *this;
	}
	else if (this->m_Row <= 1)
	{
		std::cout << "Error: <InverseRow> Size Less 2" << std::endl;
		return *this;
	}
	else
	{
		//单位矩阵 与带转换矩阵维度相同的
		Matrix uint = this->Uint();

		//结果矩阵 逆矩阵 初始状态与本矩阵相同 为不使本矩阵发生改变
		Matrix temp(this->m_Row, this->m_Col);
		Matrix* resMat = new Matrix(temp.Uint());

		//本矩阵
		Matrix localMat(*this);

		//*******************通过交换 num1*i + num2*j 实现下三角为0***************//
		for (int i = 0; i < localMat.m_Row - 1; i++)
		{

			//记录最大行所在行标
			int tempMaxRow = i;

			for (int i1 = i + 1; i1 < localMat.m_Row; i1++)
			{
				if (abs(localMat.m_Matrix[i1][i]) > abs(localMat.m_Matrix[tempMaxRow][i]))
				{
					tempMaxRow = i1;
				}
			}

			if (tempMaxRow != i)
			{
				//std::cout << i << " 行交换" << tempMaxRow << " 行" << std::endl;
				//进行交换 将当前第i行与第tempMaxRow行进行互换 初等行变换
				localMat = localMat.SwapRow(i, tempMaxRow);
				*resMat = resMat->SwapRow(i, tempMaxRow);

				//localMat.PrintMat();
			}
			

			//此对角线以下的元素通过初等变化为0
			for (int i2 = i + 1; i2 < localMat.m_Row; i2++)
			{
				if (localMat.m_Matrix[i2][i] != 0)
				{
					//std::cout << "<" << localMat.m_Matrix[i][i] << "> *" << i2 << " 行 + <" << -1.0 * (localMat.m_Matrix[i2][i]) << "> *" << i << " 行" << std::endl;

					*resMat = resMat->AddRow(i2, i, -1.0 * (localMat.m_Matrix[i2][i]) / localMat.m_Matrix[i][i]);
					localMat = localMat.AddRow(i2, i, -1.0 * (localMat.m_Matrix[i2][i]) / localMat.m_Matrix[i][i]);

					//localMat.PrintMat();

				}
			}
		}

		//错误判断
		if (localMat.m_Matrix[localMat.m_Row - 1][localMat.m_Col - 1] == 0)
		{
			std::cout << "Error: <InverseRow> marix[" << localMat.m_Row - 1 << "][" << localMat.m_Col - 1 <<"] == 0" << std::endl;
			return *this;
		}

		//*******************通过 num1*i + num2*j 实现上三角为0***************//
		for (int i = localMat.m_Row - 1; i > 0; i--)
		{
			for (int i2 = i - 1; i2 >= 0; i2--)
			{
				if (localMat.m_Matrix[i2][i] != 0)
				{
					//std::cout << "<" << localMat.m_Matrix[i][i] << "> *" << i2 << " 行 + <" << -1.0 * (localMat.m_Matrix[i2][i]) << "> *" << i << " 行" << std::endl;

					*resMat = resMat->AddRow(i2, i, -1.0 * (localMat.m_Matrix[i2][i]) / localMat.m_Matrix[i][i]);
					localMat = localMat.AddRow(i2, i, -1.0 * (localMat.m_Matrix[i2][i]) / localMat.m_Matrix[i][i]);

					//localMat.PrintMat();

				}
			}
		}

		//*******************通过 i*num 实现矩阵为单位矩阵***************//
		for (int i = 0; i < localMat.m_Row; i++)
		{
			if (localMat.m_Matrix[i][i] == 0)
			{
				std::cout << "Error: <InverseRow> matrix[" << i << "]" << "[" << i << "] == 0" << std::endl;
				return *this;
			}
			else
			{
				//std::cout << "<" << 1 / localMat.m_Matrix[i][i] << "> *" << i << " 行" << std::endl;

				*resMat = resMat->MultRow(1 / localMat.m_Matrix[i][i], i);
				localMat = localMat.MultRow(1 / localMat.m_Matrix[i][i], i);
				//localMat.PrintMat();
			}
		}
		return *resMat;
	}
}

//矩阵求逆 下三角矩阵
Matrix& Matrix::InverseDownTriangle()
{
	//错误判断 方阵检测
	if (this->m_Row != this->m_Col)
	{
		std::cout << "Error: <InverseDownTriangle> Matrix Col != Row" << std::endl;
		return *this;
	}

	//下三角求逆
	Matrix* resMat = new Matrix(*this);

	for (int i = 0; i < resMat->m_Row; i++)
	{
		for (int j = 0; j <= i; j++)
		{
			//分段求解 对角线为倒数
			if (i == j)
			{
				resMat->m_Matrix[i][j] = 1 / resMat->m_Matrix[i][j];
			}
			else
			{
				//分段求解 非对角线元素 
				double tempSum = 0.0;
				for (int k = j; k <= i - 1; k++)
				{
					tempSum += resMat->m_Matrix[i][k] * resMat->m_Matrix[k][j];
				}
				resMat->m_Matrix[i][j] = -1.0*tempSum / resMat->m_Matrix[i][i];
			}

		}
	}

	return *resMat;

}

//矩阵求逆 上三角矩阵
Matrix& Matrix::InverseUpTriangle()
{
	//错误判断 方阵检测
	if (this->m_Row != this->m_Col)
	{
		std::cout << "Error: <InverseUpTriangle> Matrix Col != Row" << std::endl;
		return *this;
	}

	//上三角求逆
	Matrix* resMat = new Matrix(*this);

	for (int j = resMat->m_Col-1; j >=0; j--)
	{
		for (int i = j; i >=0; i--)
		{
			//分段求解 对角线为倒数
			if (i == j)
			{
				resMat->m_Matrix[i][j] = 1 / resMat->m_Matrix[i][j];
			}
			else
			{
				//分段求解 非对角线元素 
				double tempSum = 0.0;
				for (int k = j; k >= i+1; k--)
				{
					tempSum += resMat->m_Matrix[i][k] * resMat->m_Matrix[k][j];
				}
				resMat->m_Matrix[i][j] = -1.0 * tempSum / resMat->m_Matrix[i][i];
			}

		}
	}

	return *resMat;
}

//矩阵LU分解 顺序分解 对于病态矩阵可能存在精度问题
void Matrix::ResolveLU(Matrix& LMat, Matrix& UMat)
{
	if (this->m_Col != this->m_Row)
	{
		std::cout << "Error: <ResolveLU> Is Not Square Matrix" << std::endl;
		return;
	}

	//存在性判定 顺序主子式不为0
	for (int i = 0; i < this->m_Row; i++)
	{
		if (this->Det(i) == 0)
		{
			std::cout << "Error: <ResolveLU> order Det = 0" << std::endl;
			return;
		}
	}

	//LU 分解
	//L矩阵为单位矩阵
	LMat = this->Uint();

	//U矩阵初始化为空矩阵
	Matrix temp(this->m_Row, this->m_Col);
	UMat = temp;

	for (int i = 0; i < this->m_Row; i++)
	{
		//计算U
		for (int j1 = i; j1 < this->m_Col; j1++)
		{
			double tempSum1 = 0.0;

			if (i != 0)
			{
				for (int j2 = 0; j2 <= i - 1; j2++)
				{
					tempSum1 += LMat.m_Matrix[i][j2] * UMat.m_Matrix[j2][j1];
				}
			}

			UMat.m_Matrix[i][j1] = this->m_Matrix[i][j1] - tempSum1;

		}

		//计算L
		for (int i1 = i; i1 < this->m_Row; i1++)
		{
			double tempSum2 = 0.0;

			if (i != 0)
			{
				for (int j2 = 0; j2 <= i - 1; j2++)
				{
					tempSum2 += LMat.m_Matrix[i1][j2] * UMat.m_Matrix[j2][i];
				}
			}

			LMat.m_Matrix[i1][i] = (this->m_Matrix[i1][i] - tempSum2)/UMat.m_Matrix[i][i];

		}
	}

}

//矩阵的LUP分解 P*A = L*U 添加了列主消元功能 
//L为主对角线元素为1的下三角矩阵 U为上二角矩阵 P为行交换矩阵 P*A=L*U
void Matrix::ResolveLUP(Matrix& LMat, Matrix& UMat, Matrix& PMat)
{
	//条件判断 矩阵行列式不为0
	if (this->Det() == 0)
	{
		std::cout << "Error: <ResolveLUP> Can't Resolve Matrix To L U P" << std::endl;
		return;
	}

	//初始化 L U P
	LMat = this->Uint();
	PMat = this->Uint();
	UMat = *this;

	//进行分解计算
	for (int i = 0; i < UMat.m_Row - 1; i++)
	{
		//记录最大行所在行标
		int tempMaxRow = i;

		for (int i1 = i + 1; i1 < UMat.m_Row; i1++)
		{
			if (abs(UMat.m_Matrix[i1][i]) > abs(UMat.m_Matrix[tempMaxRow][i]))
			{
				tempMaxRow = i1;
			}
		}

		//进行交换 将当前第i行与第tempMaxRow行进行互换 初等行变换
		UMat = UMat.SwapRow(i, tempMaxRow);

		//L矩阵做出对应交换 先交换<itempMaxRow>列再交换<itempMaxRow>行
		LMat = LMat.SwapCol(i, tempMaxRow);
		LMat = LMat.SwapRow(i, tempMaxRow);

		//P矩阵做出对应变换 交换<itempMaxRow>行
		PMat = PMat.SwapRow(i, tempMaxRow);

		//高斯消元 V矩阵消除下三角区域，L矩阵添加下三角区域
		for (int i1 = i + 1; i1 < UMat.m_Row; i1++)
		{
			//记录消元系数
			double deleteVar = UMat.m_Matrix[i1][i] / UMat.m_Matrix[i][i];

			//L矩阵列填充
			LMat.m_Matrix[i1][i] = deleteVar;

			//U矩阵列消除
			UMat = UMat.MultRow(UMat.m_Matrix[i][i], i1).AddRow(i1, i, -1.0 * UMat.m_Matrix[i1][i]).MultRow(1 / UMat.m_Matrix[i][i], i1);
		}
	}

	return;
}

测试验证：

测试代码：

int main()
{
	//定义矩阵数值
	double tempValue0[9] = {
	1.0, 2.0, 3.0,
	4.0, 5.0, 6.0,
	7.0, 8.0, 0.0
	};

	//创建矩阵
	Matrix* tempMatrix0 = new Matrix(3, 3, tempValue0);
	Matrix* tempMatrix0L = new Matrix(3, 3);
	Matrix* tempMatrix0U = new Matrix(3, 3);
	Matrix* tempMatrix0P = new Matrix(3, 3);

	//打印矩阵
	std::cout << "************************" << std::endl;
	std::cout << "矩阵求逆前:" << std::endl;
	tempMatrix0->PrintMat();

	std::cout << "矩阵求逆后:" << std::endl;
	tempMatrix0->InverseRow().PrintMat();

	std::cout << "求逆验证:" << std::endl;
	tempMatrix0->MultMat(tempMatrix0->InverseRow()).PrintMat();

	std::cout << "************************" << std::endl;
	std::cout << "矩阵LU分解前:" << std::endl;
	tempMatrix0->PrintMat();

	std::cout << "矩阵LU分解后:" << std::endl;
	tempMatrix0->ResolveLUP(*tempMatrix0L, *tempMatrix0U, *tempMatrix0P);
	std::cout << "矩阵L:" << std::endl;
	tempMatrix0L->PrintMat();
	std::cout << "矩阵U:" << std::endl;
	tempMatrix0U->PrintMat();
	std::cout << "矩阵P:" << std::endl;
	tempMatrix0P->PrintMat();

	system("pause");
	return 0;
}

应用输出：

************************
矩阵求逆前:
1.000000e+00    2.000000e+00    3.000000e+00
4.000000e+00    5.000000e+00    6.000000e+00
7.000000e+00    8.000000e+00    0.000000e+00

矩阵求逆后:
-1.777778e+00   8.888889e-01    -1.111111e-01
1.555556e+00    -7.777778e-01   2.222222e-01
-1.111111e-01   2.222222e-01    -1.111111e-01

求逆验证:
1.000000e+00    -1.110223e-16   0.000000e+00
-2.220446e-16   1.000000e+00    0.000000e+00
1.776357e-15    -8.881784e-16   1.000000e+00

************************
矩阵LU分解前:
1.000000e+00    2.000000e+00    3.000000e+00
4.000000e+00    5.000000e+00    6.000000e+00
7.000000e+00    8.000000e+00    0.000000e+00

矩阵LU分解后:
矩阵L:
1.000000e+00    0.000000e+00    0.000000e+00
1.428571e-01    1.000000e+00    0.000000e+00
5.714286e-01    5.000000e-01    1.000000e+00

矩阵U:
7.000000e+00    8.000000e+00    0.000000e+00
0.000000e+00    8.571429e-01    3.000000e+00
0.000000e+00    0.000000e+00    4.500000e+00

矩阵P:
0.000000e+00    0.000000e+00    1.000000e+00
1.000000e+00    0.000000e+00    0.000000e+00
0.000000e+00    1.000000e+00    0.000000e+00

请按任意键继续. . .

matlab验证：

>> tempMatrix0 = [1 2 3; 4 5 6; 7 8 0];
>> tempMatrix0^-1

ans =

   -1.7778    0.8889   -0.1111
    1.5556   -0.7778    0.2222
   -0.1111    0.2222   -0.1111

>> [L, U, P] = lu(tempMatrix0)

L =

    1.0000         0         0
    0.1429    1.0000         0
    0.5714    0.5000    1.0000


U =

    7.0000    8.0000         0
         0    0.8571    3.0000
         0         0    4.5000


P =

     0     0     1
     1     0     0
     0     1     0

2、private variable

私有成员变量

	double** m_Matrix;						//矩阵

	int m_Row;								//矩阵行数
	int m_Col;								//矩阵列数

3、全部源码

为了方便大家复制应用，这里直接贴出源码

Matrix.h声明文件：

#ifndef _MATRIX_H_
#define _MATRIX_H_
#include <iostream>
#include <math.h>
#include <vector>

//矩阵最大容量
#define MAX_COUNT 500
#define MIN_DET 1e-12				//行列式最小数值

class Matrix
{
public:
	//******************************构造函数与析构函数********************************//
	/*
	函数名称:		无参构造函数
	*/
	Matrix();

	/*
	函数名称:		矩阵有参构造函数，初始化为row行、col列的0矩阵
	row:			矩阵行数
	col:			矩阵列数
	*/
	Matrix(int row, int col);

	/*
	函数名称:		矩阵有参构造函数，初始化为row行、col列、数值为mat的矩阵
	row:			矩阵行数
	col:			矩阵列数
	*mat:			矩阵数值一维数组
	*/
	Matrix(int row, int col, double* mat);

	/*
	函数名称:		深拷贝构造函数
	mat:			需要复制的矩阵
	*/
	Matrix(const Matrix& mat);

	/*
	函数名称:		析构函数
	*/
	~Matrix();

	//*******************获取矩阵*****************//
	/*
	函数名称:		获取矩阵的第row行、第col列元素数值
	row:			矩阵行数
	col:			矩阵列数
	*/
	double GetMatrixEle(int row, int col);

	//*******************设置矩阵*****************//
	/*
	函数名称:		设置矩阵第row行、第col列数值
	row:			矩阵行数
	col:			矩阵列数
	value:			设置的矩阵数值
	*/
	void SetMatrixEle(int row, int col, double value);

	/*
	函数名称:		深拷贝矩阵
	mat:			需要复制的矩阵
	*/
	Matrix CopyMat(const Matrix mat);

	//********************************矩阵的相关计算**********************************//
	
	//*******************打印矩阵*****************//
	/*
	函数名称:		打印矩阵
	*/
	void PrintMat();
	
	//*****************矩阵基本操作***************//
	/*
	函数名称:		矩阵转置，返回的是自身引用，可链式调用
	*/
	Matrix& Transpose();

	/*
	函数名称:		等维度的单位矩阵，前提是方阵
	*/
	Matrix& Uint();

	//****************矩阵保留与剔除**************//
	/*
	函数名称:		剔除矩阵中以index为行标和列标的行和列，num代表index的大小
	*index:			矩阵中的行号与列号一维数组
	num:			index动态数组长度
	*/
	Matrix& DeleteMat(int *index, int num);

	/*
	函数名称:		剔除矩阵中以index为行标和列标的行和列，num代表index的大小
	*index:			矩阵中的行号与列号一维动态数组
	num:			index动态数组长度
	*/
	Matrix& DeleteMat(std::vector<int> index, int num);

	/*
	函数名称:		剔除矩阵中以index为行标的行，num代表index的大小
	*index:			矩阵中的行号一维数组
	num:			index动态数组长度
	*/
	Matrix& DeleteRow(int* index, int num);

	/*
	函数名称:		剔除矩阵中以index为行标的行，num代表index的大小
	*index:			矩阵中的行号一维动态数组
	num:			index动态数组长度
	*/
	Matrix& DeleteRow(std::vector<int> index, int num);

	/*
	函数名称:		剔除矩阵中以index为列标的列，num代表index的大小
	*index:			矩阵中的列号一维数组
	num:			index动态数组长度
	*/
	Matrix& DeleteCol(int* index, int num);

	/*
	函数名称:		剔除矩阵中以index为列标的列，num代表index的大小
	*index:			矩阵中的列号一维动态数组
	num:			index动态数组长度
	*/
	Matrix& DeleteCol(std::vector<int> index, int num);

	//******************矩阵的替换****************//
	/*
	函数名称:		替换矩阵中行标和列标为 index中的行与列，num代表index的大小, mat是需要替换的矩阵
	*index:			矩阵中的行标和列标的一维数组
	num:			index动态数组长度
	mat:			需要替换的矩阵
	*/
	Matrix& ReplaceMat(int* index, int num, Matrix& mat);

	/*
	函数名称:		替换矩阵中行标和列标为 index中的行与列，num代表index的大小, mat是需要替换的矩阵
	*index:			矩阵中的行标和列标的一维动态数组
	num:			index动态数组长度
	mat:			需要替换的矩阵
	*/
	Matrix& ReplaceMat(std::vector<int> index, int num, Matrix& mat);

	/*
	函数名称:		替换矩阵中行标为 index中的行，num代表index的大小, mat是需要替换的矩阵
	*index:			矩阵中的行标的一维数组
	num:			index动态数组长度
	mat:			需要替换的矩阵
	*/
	Matrix& ReplaceRow(int* index, int num, Matrix& mat);

	/*
	函数名称:		替换矩阵中行标为 index中的行，num代表index的大小, mat是需要替换的矩阵
	*index:			矩阵中的行标的一动态维数组
	num:			index动态数组长度
	mat:			需要替换的矩阵
	*/
	Matrix& ReplaceRow(std::vector<int> index, int num, Matrix& mat);

	/*
	函数名称:		替换矩阵中列标为 index中的列，num代表index的大小, mat是需要替换的矩阵
	*index:			矩阵中的列标的一维数组
	num:			index动态数组长度
	mat:			需要替换的矩阵
	*/
	Matrix& ReplaceCol(int* index, int num, Matrix& mat);

	/*
	函数名称:		替换矩阵中列标为 index中的列，num代表index的大小, mat是需要替换的矩阵
	*index:			矩阵中的列标的一维动态数组
	num:			index动态数组长度
	mat:			需要替换的矩阵
	*/
	Matrix& ReplaceCol(std::vector<int> index, int num, Matrix& mat);

	//*****************矩阵初等变化***************//
	/*
	函数名称:		交换矩阵中行标为row0与row1的元素
	row0:			矩阵行标0
	row1:			矩阵行标1
	*/
	Matrix& SwapRow(int row0, int row1);

	/*
	函数名称:		交换矩阵中列标为col0与col1的元素
	col0:			矩阵列标0
	col1:			矩阵列标1
	*/
	Matrix& SwapCol(int col0, int col1);

	/*
	函数名称:		矩阵行加法 rowLocal = rowLocal + rate *rowAdd
	rowLocal:		矩阵行标，被加数
	rowAdd:			矩阵行标，加数
	rate:			加数前倍数
	*/
	Matrix& AddRow(int rowLocal, int rowAdd, double rate = 1.0);

	//矩阵加法 某列 + 倍数*某列
	/*
	函数名称:		矩阵列加法 colLocal = colLocal + rate * colAdd
	colLocal:		矩阵列标，被加数
	colAdd:			矩阵列标，加数
	rate:			加数前倍数
	*/
	Matrix& AddCol(int colLocal, int colAdd, double rate = 1.0);

	//*******************矩阵加法*****************//
	/*
	函数名称:		矩阵加法 本矩阵 = 本矩阵 + mat 前提是两个矩阵维度一致
	mat:			加数矩阵
	*/
	Matrix& AddMat(Matrix& mat);
	
	//*******************矩阵乘法*****************//
	/*
	函数名称:		矩阵乘法 本矩阵 = 本矩阵*num 
	num:			矩阵乘数
	*/
	Matrix& MultNum(double num);

	/*
	函数名称:		矩阵乘法（运算符重载） 本矩阵 = 本矩阵*num 
	num:			矩阵乘数
	*/
	Matrix& operator * (double num);

	/*
	函数名称:		矩阵某行乘数值row = row*num
	num:			矩阵某列乘数
	row:			矩阵行标
	*/
	Matrix& MultRow(double num, int row);

	/*
	函数名称:		矩阵某列乘数值col = col *num
	num:			矩阵某列乘数
	col:			矩阵列标
	*/
	Matrix& MultCol(double num, int col);
	
	/*
	函数名称:		矩阵乘法，按照矩阵相乘规则
	inputMat:		乘数矩阵
	*/
	Matrix& MultMat(Matrix& inputMat);

	//******************行列式相关操作***********************//
	/*
	函数名称:		求解矩阵对应行列式数值，前提为方阵，按照定义求解，时间复杂度为O(n!*n)，一般不用此方法求解
	*/
	double Det();

	/*
	函数名称:		求解矩阵对应行列式的顺序主子式，前提为方阵，按照定义求解，时间复杂度为O(n!*n)，一般不用此方法求解
	order:			阶数
	*/
	double Det(int order);
	 
	/*
	函数名称:		矩阵行标为row、列标为col的余子式
	row:			矩阵行标
	col:			矩阵列标
	*/
	Matrix& ChildMatrix(int row, int col);

	/*
	函数名称:		通过高斯列主消元求解矩阵行列式数值，最为常用
	*/
	double DetRow();

	//*********************矩阵求逆********************//
	/*
	函数名称:		矩阵求逆，按照定义求解，1/|A|*(A*)，时间复杂度为O(n!*n)，一般不用此方法
	*/
	Matrix& Inverse();

	/*
	函数名称:		矩阵求逆，通过行初等变化，高斯列主消元法求解
	*/
	Matrix& InverseRow();

	/*
	函数名称:		矩阵求逆，只针对于下三角矩阵进行求解
	*/
	Matrix& InverseDownTriangle();

	/*
	函数名称:		矩阵求逆，只针对于上三角矩阵进行求解
	*/
	Matrix& InverseUpTriangle();

	//矩阵LU分解
	/*
	函数名称:		矩阵LU分解
	LMat:			矩阵分解后的L矩阵
	UMat:			矩阵分解后的U矩阵
	*/
	void ResolveLU(Matrix& LMat, Matrix& UMat);

	/*
	函数名称:		矩阵的LUP分解 P*A = L*U 添加了列主消元功能
	LMat:			矩阵分解后的L矩阵
	UMat:			矩阵分解后的U矩阵
	PMat:			矩阵分解后的P矩阵
	*/
	void ResolveLUP(Matrix& LMat, Matrix& UMat, Matrix& PMat);


private:

	double** m_Matrix;						//矩阵

	int m_Row;								//矩阵行数
	int m_Col;								//矩阵列数

};

#endif

Matrix.cpp实现文件：

#include "Matrix.h"

//******************************构造函数与析构函数********************************//
Matrix::Matrix()
{

}

//初始化矩阵 默认值为0
Matrix::Matrix(int row, int col)
{
	this->m_Row = row;
	this->m_Col = col;

	//开辟内存
	this->m_Matrix = new double* [row];
	for (int i = 0; i < row; i++)
	{
		this->m_Matrix[i] = new double[col] {0.0};
	}

}

//初始化矩阵 设定数值
Matrix::Matrix(int row, int col, double *mat)
{
	this->m_Row = row;
	this->m_Col = col;

	//开辟内存
	this->m_Matrix = new double* [row];
	for (int i = 0; i < row; i++)
	{
		this->m_Matrix[i] = new double[col] {0.0};
	}

	//矩阵赋值
	for(int i = 0; i<row; i++)
	{
		for (int j = 0; j < col; j++)
		{
			this->m_Matrix[i][j] = mat[i * col + j];
		}
	}
}

//深拷贝
Matrix::Matrix(const Matrix& mat)
{
	//行列传递
	this->m_Row = mat.m_Row;
	this->m_Col = mat.m_Col;

	//矩阵深拷贝
	this->m_Matrix = new double* [this->m_Row];
	for (int i = 0; i < this->m_Row; i++)
	{
		this->m_Matrix[i] = new double[this->m_Col];
		memcpy(this->m_Matrix[i], mat.m_Matrix[i], sizeof(double) * this->m_Col);
	}
}

Matrix::~Matrix()
{
	//释放矩阵每一行
	for (int i = 0; i < this->m_Row; i++)
	{
		if (this->m_Matrix[i] != NULL)
		{
			delete[]this->m_Matrix[i];
			this->m_Matrix[i] = NULL;
		}
	}

	//释放矩阵顶点
	if (this->m_Matrix != NULL)
	{
		delete[]this->m_Matrix;
		this->m_Matrix = NULL;
	}
}
//获取矩阵某个元素 某行某列
double Matrix::GetMatrixEle(int row, int col)
{
	if (row >= this->m_Row)
	{
		std::cout << "Error: <GetMatrixEle> Input row >= m_Row" << std::endl;
		return 0.0;
	}
	else if (col >= this->m_Col)
	{
		std::cout << "Error: <GetMatrixEle> Input col >= m_Col" << std::endl;
		return 0.0;
	}
	else
	{
		return this->m_Matrix[row][col];
	}
}

//*******************设置矩阵*****************//
void Matrix::SetMatrixEle(int row, int col, double value)
{
	if (row >= this->m_Row)
	{
		std::cout << "Error: <SetMatrixEle> Input row >= m_Row" << std::endl;
		return;
	}
	else if (col >= this->m_Col)
	{
		std::cout << "Error: <SetMatrixEle> Input col >= m_Col" << std::endl;
		return;
	}
	else
	{
		this->m_Matrix[row][col] = value;
		return;
	}
}

Matrix Matrix::CopyMat(const Matrix mat)
{
	//行列传递
	this->m_Row = mat.m_Row;
	this->m_Col = mat.m_Col;

	//矩阵深拷贝
	this->m_Matrix = new double* [this->m_Row];
	for (int i = 0; i < this->m_Row; i++)
	{
		this->m_Matrix[i] = new double[this->m_Col];
		memcpy(this->m_Matrix[i], mat.m_Matrix[i], sizeof(double) * this->m_Col);
	}

	return *this;
}

//*******************打印矩阵*****************//
//矩阵输出
void Matrix::PrintMat()
{
	for (int i = 0; i < this->m_Row; i++)
	{
		for (int j = 0; j < this->m_Col; j++)
		{
			std::cout.setf(std::ios::scientific);		//科学计数法表示
			std::cout << this->m_Matrix[i][j] << "\t";
		}
		std::cout << std::endl;
	}
	std::cout << std::endl;
}

//*****************矩阵基本操作***************//
//矩阵转置
Matrix& Matrix::Transpose()
{
	Matrix* resMat = new Matrix(this->m_Col, this->m_Row);

	for (int i = 0; i < this->m_Row; i++)
	{
		for (int j = 0; j < this->m_Col; j++)
		{
			resMat->m_Matrix[j][i] = this->m_Matrix[i][j];
		}
	}

	return *resMat;
}

//求等长度单位矩阵
Matrix& Matrix::Uint()
{
	//矩阵是否为方阵
	if (this->m_Col != this->m_Row)
	{
		std::cout << "Error: <Uint> Row != Col" << std::endl;

		Matrix* resMat = new Matrix(this->m_Row, this->m_Row);
		return *resMat;
	}
	else
	{
		//单位矩阵初始化
		Matrix* resMat = new Matrix(this->m_Row, this->m_Col);

		//单位矩阵生成
		for (int i = 0; i < this->m_Row; i++)
		{
			resMat->m_Matrix[i][i] = 1.0;
		}

		return *resMat;
	}
}

//****************矩阵保留与剔除**************//
//剔除矩阵的 index中的行与列，num代表index的大小
Matrix& Matrix::DeleteMat(int* index, int num)
{
	//结果矩阵
	Matrix* resMat = new Matrix(this->m_Row - num, this->m_Col - num);

	int recIndex[MAX_COUNT];
	int currIndex = 0;

	//检验数据有效性
	for (int i = 0; i < num; i++)
	{
		//越界判定
		if (index[i] >= this->m_Row)
		{
			std::cout << "Error: <DeleteMat> Input index[" << i << "] = " << index[i] << " >= m_Row" << std::endl;
			return *this;
		}
		else if (index[i] >= this->m_Col)
		{
			std::cout << "Error: <DeleteMat> Input index[" << i << "] = " << index[i] << " >= m_Col" << std::endl;
			return *this;
		}
	}

	//筛选出剔除后行数
	for (int iRow = 0; iRow < this->m_Row; iRow++)
	{
		for (int iNum = 0; iNum < num; iNum++)
		{
			if (iRow == index[iNum])
			{
				break;
			}

			if (iNum == num-1)
			{
				recIndex[currIndex++] = iRow;
			}
		}
	}

	//加入元素
	for (int iRow = 0; iRow < resMat->m_Row; iRow++)
	{
		for (int iCol = 0; iCol < resMat->m_Col; iCol++)
		{
			resMat->m_Matrix[iRow][iCol] = this->m_Matrix[recIndex[iRow]][recIndex[iCol]];
		}
	}

	return *resMat;
	
}

Matrix& Matrix::DeleteMat(std::vector<int> index, int num)
{
	//结果矩阵
	Matrix* resMat = new Matrix(this->m_Row - num, this->m_Col - num);

	int recIndex[MAX_COUNT];
	int currIndex = 0;

	//检验数据有效性
	for (int i = 0; i < num; i++)
	{
		//越界判定
		if (index[i] >= this->m_Row)
		{
			std::cout << "Error: <DeleteMat> Input index[" << i << "] = " << index[i] << " >= m_Row" << std::endl;
			return *this;
		}
		else if (index[i] >= this->m_Col)
		{
			std::cout << "Error: <DeleteMat> Input index[" << i << "] = " << index[i] << " >= m_Col" << std::endl;
			return *this;
		}
	}

	//筛选出剔除后行数
	for (int iRow = 0; iRow < this->m_Row; iRow++)
	{
		for (int iNum = 0; iNum < num; iNum++)
		{
			if (iRow == index[iNum])
			{
				break;
			}

			if (iNum == num - 1)
			{
				recIndex[currIndex++] = iRow;
			}
		}
	}

	//加入元素
	for (int iRow = 0; iRow < resMat->m_Row; iRow++)
	{
		for (int iCol = 0; iCol < resMat->m_Col; iCol++)
		{
			resMat->m_Matrix[iRow][iCol] = this->m_Matrix[recIndex[iRow]][recIndex[iCol]];
		}
	}

	return *resMat;
}

//剔除矩阵的 index中的行，num代表index的大小
Matrix& Matrix::DeleteRow(int* index, int num)
{
	//结果矩阵
	Matrix* resMat = new Matrix(this->m_Row - num, this->m_Col);

	int recIndex[MAX_COUNT];
	int currIndex = 0;

	//检验数据有效性
	for (int i = 0; i < num; i++)
	{
		//越界判定
		if (index[i] >= this->m_Row)
		{
			std::cout << "Error: <DeleteMat> Input index[" << i << "] = " << index[i] << " >= m_Row" << std::endl;
			return *this;
		}
	}

	//筛选出剔除后行数
	for (int iRow = 0; iRow < this->m_Row; iRow++)
	{
		for (int iNum = 0; iNum < num; iNum++)
		{
			if (iRow == index[iNum])
			{
				break;
			}

			if (iNum == num - 1)
			{
				recIndex[currIndex++] = iRow;
			}
		}
	}

	//加入元素
	for (int iRow = 0; iRow < resMat->m_Row; iRow++)
	{
		for (int iCol = 0; iCol < resMat->m_Col; iCol++)
		{
			resMat->m_Matrix[iRow][iCol] = this->m_Matrix[recIndex[iRow]][iCol];
		}
	}

	return *resMat;
}

Matrix& Matrix::DeleteRow(std::vector<int> index, int num)
{
	//结果矩阵
	Matrix* resMat = new Matrix(this->m_Row - num, this->m_Col);

	int recIndex[MAX_COUNT];
	int currIndex = 0;

	//检验数据有效性
	for (int i = 0; i < num; i++)
	{
		//越界判定
		if (index[i] >= this->m_Row)
		{
			std::cout << "Error: <DeleteMat> Input index[" << i << "] = " << index[i] << " >= m_Row" << std::endl;
			return *this;
		}
	}

	//筛选出剔除后行数
	for (int iRow = 0; iRow < this->m_Row; iRow++)
	{
		for (int iNum = 0; iNum < num; iNum++)
		{
			if (iRow == index[iNum])
			{
				break;
			}

			if (iNum == num - 1)
			{
				recIndex[currIndex++] = iRow;
			}
		}
	}

	//加入元素
	for (int iRow = 0; iRow < resMat->m_Row; iRow++)
	{
		for (int iCol = 0; iCol < resMat->m_Col; iCol++)
		{
			resMat->m_Matrix[iRow][iCol] = this->m_Matrix[recIndex[iRow]][iCol];
		}
	}

	return *resMat;
}

Matrix& Matrix::DeleteCol(int* index, int num)
{
	//结果矩阵
	Matrix* resMat = new Matrix(this->m_Row, this->m_Col - num);

	int recIndex[MAX_COUNT];
	int currIndex = 0;

	//检验数据有效性
	for (int i = 0; i < num; i++)
	{
		//越界判定
		if (index[i] >= this->m_Row)
		{
			std::cout << "Error: <DeleteMat> Input index[" << i << "] = " << index[i] << " >= m_Row" << std::endl;
			return *this;
		}
	}

	//筛选出剔除后行数
	for (int iRow = 0; iRow < this->m_Row; iRow++)
	{
		for (int iNum = 0; iNum < num; iNum++)
		{
			if (iRow == index[iNum])
			{
				break;
			}

			if (iNum == num - 1)
			{
				recIndex[currIndex++] = iRow;
			}
		}
	}

	//加入元素
	for (int iRow = 0; iRow < resMat->m_Row; iRow++)
	{
		for (int iCol = 0; iCol < resMat->m_Col; iCol++)
		{
			resMat->m_Matrix[iRow][iCol] = this->m_Matrix[iRow][recIndex[iCol]];
		}
	}

	return *resMat;
}

Matrix& Matrix::DeleteCol(std::vector<int> index, int num)
{
	//结果矩阵
	Matrix* resMat = new Matrix(this->m_Row, this->m_Col - num);

	int recIndex[MAX_COUNT];
	int currIndex = 0;

	//检验数据有效性
	for (int i = 0; i < num; i++)
	{
		//越界判定
		if (index[i] >= this->m_Row)
		{
			std::cout << "Error: <DeleteMat> Input index[" << i << "] = " << index[i] << " >= m_Row" << std::endl;
			return *this;
		}
	}

	//筛选出剔除后行数
	for (int iRow = 0; iRow < this->m_Row; iRow++)
	{
		for (int iNum = 0; iNum < num; iNum++)
		{
			if (iRow == index[iNum])
			{
				break;
			}

			if (iNum == num - 1)
			{
				recIndex[currIndex++] = iRow;
			}
		}
	}

	//加入元素
	for (int iRow = 0; iRow < resMat->m_Row; iRow++)
	{
		for (int iCol = 0; iCol < resMat->m_Col; iCol++)
		{
			resMat->m_Matrix[iRow][iCol] = this->m_Matrix[iRow][recIndex[iCol]];
		}
	}

	return *resMat;
}

//******************矩阵的替换****************//
//替换矩阵中的行和列 index中的行与列，num代表index的大小
Matrix& Matrix::ReplaceMat(int* index, int num, Matrix& mat)
{

	//错误判定 方阵
	if (this->m_Row != this->m_Col)
	{
		std::cout << "Error: <ReplaceMat> this m_Col != m_Row" << std::endl;
		return *this;
	}

	//检验插入矩阵为方阵
	if (mat.m_Row != mat.m_Col)
	{
		std::cout << "Error: <ReplaceMat> mat m_Col != m_Row" << std::endl;
		return *this;
	}

	//检验插入矩阵大小与num保持一致
	if (mat.m_Col != num)
	{
		std::cout << "Error: <ReplaceMat> num != mat.m_Col" << std::endl;
		return *this;
	}

	//检验数据有效性
	for (int i = 0; i < num; i++)
	{
		//越界判定
		if (index[i] >= this->m_Row)
		{
			std::cout << "Error: <ReplaceMat> Input index[" << i << "] = " << index[i] << " >= m_Row" << std::endl;
			return *this;
		}
		else if (index[i] >= this->m_Col)
		{
			std::cout << "Error: <ReplaceMat> Input index[" << i << "] = " << index[i] << " >= m_Col" << std::endl;
			return *this;
		}
	}

	//结果矩阵
	Matrix* resMat = new Matrix(*this);

	//加入元素
	for (int iRow = 0; iRow < num; iRow++)
	{
		for (int iCol = 0; iCol < num; iCol++)
		{
			resMat->m_Matrix[index[iRow]][index[iCol]] = mat.m_Matrix[iRow][iCol];
		}
	}

	return *resMat;
}

Matrix& Matrix::ReplaceMat(std::vector<int> index, int num, Matrix& mat)
{
	//错误判定 方阵
	if (this->m_Row != this->m_Col)
	{
		std::cout << "Error: <ReplaceMat> this m_Col != m_Row" << std::endl;
		return *this;
	}

	//检验插入矩阵为方阵
	if (mat.m_Row != mat.m_Col)
	{
		std::cout << "Error: <ReplaceMat> mat m_Col != m_Row" << std::endl;
		return *this;
	}

	//检验插入矩阵大小与num保持一致
	if (mat.m_Col != num)
	{
		std::cout << "Error: <ReplaceMat> num != mat.m_Col" << std::endl;
		return *this;
	}

	//检验数据有效性
	for (int i = 0; i < num; i++)
	{
		//越界判定
		if (index[i] >= this->m_Row)
		{
			std::cout << "Error: <ReplaceMat> Input index[" << i << "] = " << index[i] << " >= m_Row" << std::endl;
			return *this;
		}
		else if (index[i] >= this->m_Col)
		{
			std::cout << "Error: <ReplaceMat> Input index[" << i << "] = " << index[i] << " >= m_Col" << std::endl;
			return *this;
		}
	}

	//结果矩阵
	Matrix* resMat = new Matrix(*this);

	//加入元素
	for (int iRow = 0; iRow < num; iRow++)
	{
		for (int iCol = 0; iCol < num; iCol++)
		{
			resMat->m_Matrix[index[iRow]][index[iCol]] = mat.m_Matrix[iRow][iCol];
		}
	}

	return *resMat;
}

//替换矩阵中的行 index中的行，num代表index的大小, mat是需要替换的矩阵
Matrix& Matrix::ReplaceRow(int* index, int num, Matrix& mat)
{
	//检验插入矩阵大小与num保持一致
	if (mat.m_Row != num)
	{
		std::cout << "Error: <ReplaceRow> num != mat.m_Row" << std::endl;
		return *this;
	}

	//检验数据有效性
	for (int i = 0; i < num; i++)
	{
		//越界判定
		if (index[i] >= this->m_Row)
		{
			std::cout << "Error: <ReplaceRow> Input index[" << i << "] = " << index[i] << " >= m_Row" << std::endl;
			return *this;
		}
	}

	//当前矩阵列数应与mat列数一致
	if (this->m_Col != mat.m_Col)
	{
		std::cout << "Error: <ReplaceRow> this->m_Col != mat.m_Col" << std::endl;
		return *this;
	}

	//结果矩阵
	Matrix* resMat = new Matrix(*this);

	//加入元素
	for (int iRow = 0; iRow < num; iRow++)
	{
		for (int iCol = 0; iCol < resMat->m_Col; iCol++)
		{
			resMat->m_Matrix[index[iRow]][iCol] = mat.m_Matrix[iRow][iCol];
		}
	}

	return *resMat;
}

Matrix& Matrix::ReplaceRow(std::vector<int> index, int num, Matrix& mat)
{
	//检验插入矩阵大小与num保持一致
	if (mat.m_Row != num)
	{
		std::cout << "Error: <ReplaceRow> num != mat.m_Row" << std::endl;
		return *this;
	}

	//检验数据有效性
	for (int i = 0; i < num; i++)
	{
		//越界判定
		if (index[i] >= this->m_Row)
		{
			std::cout << "Error: <ReplaceRow> Input index[" << i << "] = " << index[i] << " >= m_Row" << std::endl;
			return *this;
		}
	}

	//当前矩阵列数应与mat列数一致
	if (this->m_Col != mat.m_Col)
	{
		std::cout << "Error: <ReplaceRow> this->m_Col != mat.m_Col" << std::endl;
		return *this;
	}

	//结果矩阵
	Matrix* resMat = new Matrix(*this);

	//加入元素
	for (int iRow = 0; iRow < num; iRow++)
	{
		for (int iCol = 0; iCol < resMat->m_Col; iCol++)
		{
			resMat->m_Matrix[index[iRow]][iCol] = mat.m_Matrix[iRow][iCol];
		}
	}

	return *resMat;
}

//替换矩阵中的列 index中的列，num代表index的大小, mat是需要替换的矩阵
Matrix& Matrix::ReplaceCol(int* index, int num, Matrix& mat)
{
	//检验插入矩阵大小与num保持一致
	if (mat.m_Col != num)
	{
		std::cout << "Error: <ReplaceCol> mat.m_Col != num" << std::endl;
		return *this;
	}

	//检验数据有效性
	for (int i = 0; i < num; i++)
	{
		//越界判定
		if (index[i] >= this->m_Col)
		{
			std::cout << "Error: <ReplaceCol> Input index[" << i << "] = " << index[i] << " >= m_Col" << std::endl;
			return *this;
		}
	}

	//当前矩阵行数应与mat行数一致
	if (this->m_Row != mat.m_Row)
	{
		std::cout << "Error: <ReplaceCol> this->m_Row != mat.m_Row" << std::endl;
		return *this;
	}

	//结果矩阵
	Matrix* resMat = new Matrix(*this);

	//加入元素
	for (int iRow = 0; iRow < resMat->m_Row; iRow++)
	{
		for (int iCol = 0; iCol < num; iCol++)
		{
			resMat->m_Matrix[iRow][index[iCol]] = mat.m_Matrix[iRow][iCol];
		}
	}

	return *resMat;
}

Matrix& Matrix::ReplaceCol(std::vector<int> index, int num, Matrix& mat)
{
	//检验插入矩阵大小与num保持一致
	if (mat.m_Col != num)
	{
		std::cout << "Error: <ReplaceCol> mat.m_Col != num" << std::endl;
		return *this;
	}

	//检验数据有效性
	for (int i = 0; i < num; i++)
	{
		//越界判定
		if (index[i] >= this->m_Col)
		{
			std::cout << "Error: <ReplaceCol> Input index[" << i << "] = " << index[i] << " >= m_Col" << std::endl;
			return *this;
		}
	}

	//当前矩阵行数应与mat行数一致
	if (this->m_Row != mat.m_Row)
	{
		std::cout << "Error: <ReplaceCol> this->m_Row != mat.m_Row" << std::endl;
		return *this;
	}

	//结果矩阵
	Matrix* resMat = new Matrix(*this);

	//加入元素
	for (int iRow = 0; iRow < resMat->m_Row; iRow++)
	{
		for (int iCol = 0; iCol < num; iCol++)
		{
			resMat->m_Matrix[iRow][index[iCol]] = mat.m_Matrix[iRow][iCol];
		}
	}

	return *resMat;
}

//*****************矩阵初等变化***************//
Matrix& Matrix::SwapRow(int row0, int row1)
{
	//错误判定 越界
	if ((this->m_Row <= row0) || (this->m_Col <= row1))
	{
		std::cout << "Error: <SwapRow> Input row0 Or row1 More Than m_Row" << std::endl;
		return *this;
	}
	else if ((0 > row0) || (0 > row1))
	{
		std::cout << "Error: <SwapRow> Input row0 Or row1 Less 0" << std::endl;
		return *this;
	}
	else
	{
		//结果矩阵初始化
		Matrix* resMat = new Matrix(*this);

		//中转临时变量
		double temp = 0.0;

		for (int j = 0; j < resMat->m_Col; j++)
		{
			temp = resMat->m_Matrix[row0][j];
			resMat->m_Matrix[row0][j] = resMat->m_Matrix[row1][j];
			resMat->m_Matrix[row1][j] = temp;
		}

		return*resMat;
	}
}

Matrix& Matrix::SwapCol(int col0, int col1)
{
	//错误判定 越界
	if ((this->m_Col <= col0) || (this->m_Col <= col1))
	{
		std::cout << "Error: <SwapCol> Input col0 Or col1 More Than m_Col" << std::endl;
		return *this;
	}
	else if ((0 > col0) || (0 > col1))
	{
		std::cout << "Error: <SwapCol> Input col0 Or col1 Less 0" << std::endl;
		return *this;
	}
	else
	{
		//结果矩阵初始化
		Matrix* resMat = new Matrix(*this);

		//中转临时变量
		double temp = 0.0;

		for (int i = 0; i < resMat->m_Row; i++)
		{
			temp = resMat->m_Matrix[i][col0];
			resMat->m_Matrix[i][col0] = resMat->m_Matrix[i][col1];
			resMat->m_Matrix[i][col1] = temp;
		}

		return*resMat;
	}
}

//矩阵加法 某行 + 倍数*某行
Matrix& Matrix::AddRow(int rowLocal, int rowAdd, double rate)
{
	if ((this->m_Row <= rowLocal) || (this->m_Row <= rowAdd))
	{
		std::cout << "Error: <AddRow> Input rowLocal Or rowAdd More Than m_Row" << std::endl;
		return *this;
	}
	else if ((0 > rowLocal) || (0 > rowAdd))
	{
		std::cout << "Error: <AddRow> Input rowLocal Or rowAdd Less 0" << std::endl;
		return *this;
	}
	else
	{
		//结果矩阵初始化
		Matrix* resMat = new Matrix(*this);

		//指定行相加
		for (int j = 0; j < resMat->m_Col; j++)
		{
			resMat->m_Matrix[rowLocal][j] += rate * resMat->m_Matrix[rowAdd][j];
		}

		return *resMat;
	}
}

//矩阵加法 某列 + 倍数*某列
Matrix& Matrix::AddCol(int colLocal, int colAdd, double rate)
{
	if ((this->m_Col <= colLocal) || (this->m_Col <= colAdd))
	{
		std::cout << "Error: <AddCol> Input colLocal Or colAdd More Than m_Col" << std::endl;
		return *this;
	}
	else if ((0 > colLocal) || (0 > colAdd))
	{
		std::cout << "Error: <AddCol> Input colLocal Or colAdd Less 0" << std::endl;
		return *this;
	}
	else
	{
		//结果矩阵初始化
		Matrix* resMat = new Matrix(*this);

		//指定列相加
		for (int i = 0; i < resMat->m_Row; i++)
		{
			resMat->m_Matrix[i][colLocal] += rate * resMat->m_Matrix[i][colAdd];
		}

		return *resMat;
	}
}

//*******************矩阵加法*****************//
Matrix& Matrix::AddMat(Matrix& mat)
{
	Matrix* ResMat = new Matrix(*this);

	for (int i = 0; i < ResMat->m_Row; i++)
	{
		for (int j = 0; j < ResMat->m_Col; j++)
		{
			ResMat->m_Matrix[i][j] += mat.m_Matrix[i][j];
		}
	}

	return *ResMat;
}

//*******************矩阵乘法*****************//
//矩阵数乘
Matrix& Matrix::MultNum(double num)
{
	//结果矩阵初始化
	Matrix* resMat = new Matrix(this->m_Row, this->m_Col);

	//乘后矩阵生成
	for (int i = 0; i < this->m_Row; i++)
	{
		for (int j = 0; j < this->m_Col; j++)
		{
			resMat->m_Matrix[i][j] = num * this->m_Matrix[i][j];
		}
	}

	return *resMat;
}

//运算符重载 矩阵数乘
Matrix& Matrix::operator*(double num)
{
	//结果矩阵初始化
	Matrix* resMat = new Matrix(this->m_Row, this->m_Col);

	//乘后矩阵生成
	for (int i = 0; i < this->m_Row; i++)
	{
		for (int j = 0; j < this->m_Col; j++)
		{
			resMat->m_Matrix[i][j] = num * this->m_Matrix[i][j];
		}
	}

	return *resMat;
}

//矩阵某行乘数值 行标从0开始计数
Matrix& Matrix::MultRow(double num, int row)
{
	if (this->m_Row <= row)
	{
		std::cout << "Error: <MultRow> Input row More Than m_Row" << std::endl;
		return *this;
	}
	else if (0 > row)
	{
		std::cout << "Error: <MultRow> Input row Less 0" << std::endl;
		return *this;
	}
	else
	{
		//结果矩阵初始化
		Matrix* resMat = new Matrix(*this);

		//乘后矩阵生成
		for (int j = 0; j < this->m_Col; j++)
		{
			resMat->m_Matrix[row][j] = num * this->m_Matrix[row][j];
		}

		return *resMat;
	}

}

//矩阵某列乘数值 列标从0开始计数
Matrix& Matrix::MultCol(double num, int col)
{
	if (this->m_Col <= col)
	{
		std::cout << "Error: <MultCol> Input col More Than m_Row" << std::endl;
		return *this;
	}
	else if (0 > col)
	{
		std::cout << "Error: <MultCol> Input col Less 0" << std::endl;
		return *this;
	}
	else
	{
		//结果矩阵初始化
		Matrix* resMat = new Matrix(*this);

		//乘后矩阵生成
		for (int i = 0; i < this->m_Row; i++)
		{
			resMat->m_Matrix[i][col] = num * this->m_Matrix[i][col];
		}

		return *resMat;
	}
}



//矩阵相乘
Matrix& Matrix::MultMat(Matrix& inputMat)
{
	Matrix *resMat = new Matrix(this->m_Row, inputMat.m_Col);

	if (this->m_Col != inputMat.m_Row)
	{
		std::cout << "Matrix Mult Error!" << std::endl;
		return *resMat;
	}
	else
	{
		for (int i = 0; i < this->m_Row; i++)
		{
			for (int j = 0; j < inputMat.m_Col; j++)
			{
				for (int k = 0; k < this->m_Col; k++)
				{
					resMat->m_Matrix[i][j] += this->m_Matrix[i][k] * inputMat.m_Matrix[k][j];
				}
			}
		}

		return *resMat;
	}
}


//矩阵的行列式数值
double Matrix::Det()
{
	double res = 0.0;
	int sign = 1;

	if (this->m_Row != this->m_Col)
	{
		//错误判定
		std::cout << "Error: <Det> Matrix Col != Row" << std::endl;
		return 0;
	}
	else if (this->m_Row <= 1)
	{
		//程序终止出口
		return this->m_Matrix[0][0];
	}
	else
	{
		for (int i = 0; i < this->m_Col; i++)
		{
			Matrix* temp = &(this->ChildMatrix(0, i));
			res += sign * this->m_Matrix[0][i] * (temp->Det());
			sign = -1*sign;
			delete temp;
		}
	}

}

//矩阵行列式顺序主子式 order阶数
double Matrix::Det(int order)
{
	if (this->m_Row != this->m_Col)
	{
		//错误判定
		std::cout << "Error: <Det> Matrix Col != Row" << std::endl;
		return 0;
	}
	else if (order < 0)
	{
		std::cout << "Error: <Det>  Input Order Less 0" << std::endl;
		return 0;
	}
	else if (order >= this->m_Row)
	{
		std::cout << "Error: <Det> Input Order More Than Row" << std::endl;
		return 0;
	}
	else
	{
		Matrix tempMat(order + 1, order + 1);
		for (int i = 0; i < tempMat.m_Col; i++)
		{
			for (int j = 0; j < tempMat.m_Row; j++)
			{
				tempMat.m_Matrix[i][j] = this->m_Matrix[i][j];
			}
		}
		return tempMat.Det();
	}
}

//求解余子式
Matrix& Matrix::ChildMatrix(int row, int col)
{
	if (this->m_Row != this->m_Col)
	{
		std::cout << "Error: <ChildMatrix> Matrix row != col" << std::endl;
		return *this;
	}
	else if (this->m_Row <= 1)
	{
		std::cout << "Error: <ChildMatrix> Matrix Row Less 1 " << std::endl;
		return *this;
	}
	else if ((row > this->m_Row) || (col > this->m_Col))
	{
		std::cout << "Error: <ChildMatrix> Input Row Or Col More Than Matix Max Row Or Col" << std::endl; 
		return* this; 
	}
	else
	{
		Matrix* resMat = new Matrix(this->m_Row-1, this->m_Col-1);

		for (int i = 0; i < this->m_Row; i++)
		{
			for (int j = 0; j < this->m_Col; j++)
			{
				if ((i < row) && (j < col))
					resMat->m_Matrix[i][j] = this->m_Matrix[i][j];
				else if((i > row) && (j < col))
					resMat->m_Matrix[i-1][j] = this->m_Matrix[i][j];
				else if((i < row) && (j > col))
					resMat->m_Matrix[i][j - 1] = this->m_Matrix[i][j];
				else if((i > row) && (j > col))
					resMat->m_Matrix[i - 1][j - 1] = this->m_Matrix[i][j];
			}
		}

		return *resMat;
	}
}

//列主消元处理为上三角矩阵
double Matrix::DetRow()
{
	//交换标志位 1代表偶数次交换 -1代表奇数次交换
	int flagShift = 1;

	//本矩阵
	Matrix *localMat = new Matrix(*this);

	//行列式数值
	double resDet = 1.0;

	//*******************通过交换 num1*i + num2*j 实现下三角为0***************//
	for (int i = 0; i < localMat->m_Row - 1; i++)
	{
		//记录最大行所在行标
		int tempMaxRow = i;

		for (int i1 = i + 1; i1 < localMat->m_Row; i1++)
		{
			if (abs(localMat->m_Matrix[i1][i]) > abs(localMat->m_Matrix[tempMaxRow][i]))
			{
				tempMaxRow = i1;
			}
		}

		if (tempMaxRow != i)
		{
			//std::cout << i << " 行交换" << tempMaxRow << " 行" << std::endl;
			//进行交换 将当前第i行与第tempMaxRow行进行互换 初等行变换
	
			*localMat = localMat->SwapRow(i, tempMaxRow);

			//记录交换次数
			flagShift = -flagShift;

			//localMat->PrintMat();
		}


		//此对角线以下的元素通过初等变化为0
		for (int i2 = i + 1; i2 < localMat->m_Row; i2++)
		{
			if (localMat->m_Matrix[i2][i] != 0)
			{
				//std::cout << "<" << localMat->m_Matrix[i][i] << "> *" << i2 << " 行 + <" << -1.0 * (localMat->m_Matrix[i2][i]) << "> *" << i << " 行" << std::endl;

				*localMat = localMat->AddRow(i2, i, -1.0 * (localMat->m_Matrix[i2][i]) / localMat->m_Matrix[i][i]);

				//localMat->PrintMat();

			}
		}
	}

	//计算行列式数值 对角线相乘
	for (int i = 0; i < localMat->m_Row; i++)
	{
		resDet = resDet * localMat->m_Matrix[i][i];
	}

	//矩阵交换一次就会变号
	resDet = flagShift * resDet;

	//清理localMatrix
	delete localMat;

	return resDet;
}

//矩阵求逆
Matrix& Matrix::Inverse()
{
	if (abs(this->DetRow()) < MIN_DET)
	{
		std::cout << "Error: <Inverse> Matrix Det Near 0" << std::endl;
		return *this;
	}
	else
	{
		Matrix* resMat = new Matrix(this->m_Row, this->m_Col);
		for (int i = 0; i < this->m_Row; i++)
		{
			for (int j = 0; j < this->m_Col; j++)
			{
				Matrix* temp = &(this->ChildMatrix(j, i));
				resMat->m_Matrix[i][j] = pow(-1.0, (i + j)) / this->DetRow() * (temp->DetRow());
				delete temp;
			}
		}

		return *resMat;
	}
}

//矩阵求逆 行初等变化
Matrix& Matrix::InverseRow()
{
	//错误判断
	if (abs(this->DetRow()) < MIN_DET)
	{
		std::cout << "Error: <InverseRow> Matrix Det Near 0" << std::endl;
		return *this;
	}
	else if (this->m_Row <= 1)
	{
		std::cout << "Error: <InverseRow> Size Less 2" << std::endl;
		return *this;
	}
	else
	{
		//单位矩阵 与带转换矩阵维度相同的
		Matrix uint = this->Uint();

		//结果矩阵 逆矩阵 初始状态与本矩阵相同 为不使本矩阵发生改变
		Matrix temp(this->m_Row, this->m_Col);
		Matrix* resMat = new Matrix(temp.Uint());

		//本矩阵
		Matrix localMat(*this);

		//*******************通过交换 num1*i + num2*j 实现下三角为0***************//
		for (int i = 0; i < localMat.m_Row - 1; i++)
		{

			//记录最大行所在行标
			int tempMaxRow = i;

			for (int i1 = i + 1; i1 < localMat.m_Row; i1++)
			{
				if (abs(localMat.m_Matrix[i1][i]) > abs(localMat.m_Matrix[tempMaxRow][i]))
				{
					tempMaxRow = i1;
				}
			}

			if (tempMaxRow != i)
			{
				//std::cout << i << " 行交换" << tempMaxRow << " 行" << std::endl;
				//进行交换 将当前第i行与第tempMaxRow行进行互换 初等行变换
				localMat = localMat.SwapRow(i, tempMaxRow);
				*resMat = resMat->SwapRow(i, tempMaxRow);

				//localMat.PrintMat();
			}
			

			//此对角线以下的元素通过初等变化为0
			for (int i2 = i + 1; i2 < localMat.m_Row; i2++)
			{
				if (localMat.m_Matrix[i2][i] != 0)
				{
					//std::cout << "<" << localMat.m_Matrix[i][i] << "> *" << i2 << " 行 + <" << -1.0 * (localMat.m_Matrix[i2][i]) << "> *" << i << " 行" << std::endl;

					*resMat = resMat->AddRow(i2, i, -1.0 * (localMat.m_Matrix[i2][i]) / localMat.m_Matrix[i][i]);
					localMat = localMat.AddRow(i2, i, -1.0 * (localMat.m_Matrix[i2][i]) / localMat.m_Matrix[i][i]);

					//localMat.PrintMat();

				}
			}
		}

		//错误判断
		if (localMat.m_Matrix[localMat.m_Row - 1][localMat.m_Col - 1] == 0)
		{
			std::cout << "Error: <InverseRow> marix[" << localMat.m_Row - 1 << "][" << localMat.m_Col - 1 <<"] == 0" << std::endl;
			return *this;
		}

		//*******************通过 num1*i + num2*j 实现上三角为0***************//
		for (int i = localMat.m_Row - 1; i > 0; i--)
		{
			for (int i2 = i - 1; i2 >= 0; i2--)
			{
				if (localMat.m_Matrix[i2][i] != 0)
				{
					//std::cout << "<" << localMat.m_Matrix[i][i] << "> *" << i2 << " 行 + <" << -1.0 * (localMat.m_Matrix[i2][i]) << "> *" << i << " 行" << std::endl;

					*resMat = resMat->AddRow(i2, i, -1.0 * (localMat.m_Matrix[i2][i]) / localMat.m_Matrix[i][i]);
					localMat = localMat.AddRow(i2, i, -1.0 * (localMat.m_Matrix[i2][i]) / localMat.m_Matrix[i][i]);

					//localMat.PrintMat();

				}
			}
		}

		//*******************通过 i*num 实现矩阵为单位矩阵***************//
		for (int i = 0; i < localMat.m_Row; i++)
		{
			if (localMat.m_Matrix[i][i] == 0)
			{
				std::cout << "Error: <InverseRow> matrix[" << i << "]" << "[" << i << "] == 0" << std::endl;
				return *this;
			}
			else
			{
				//std::cout << "<" << 1 / localMat.m_Matrix[i][i] << "> *" << i << " 行" << std::endl;

				*resMat = resMat->MultRow(1 / localMat.m_Matrix[i][i], i);
				localMat = localMat.MultRow(1 / localMat.m_Matrix[i][i], i);
				//localMat.PrintMat();
			}
		}
		return *resMat;
	}
}

//矩阵求逆 下三角矩阵
Matrix& Matrix::InverseDownTriangle()
{
	//错误判断 方阵检测
	if (this->m_Row != this->m_Col)
	{
		std::cout << "Error: <InverseDownTriangle> Matrix Col != Row" << std::endl;
		return *this;
	}

	//下三角求逆
	Matrix* resMat = new Matrix(*this);

	for (int i = 0; i < resMat->m_Row; i++)
	{
		for (int j = 0; j <= i; j++)
		{
			//分段求解 对角线为倒数
			if (i == j)
			{
				resMat->m_Matrix[i][j] = 1 / resMat->m_Matrix[i][j];
			}
			else
			{
				//分段求解 非对角线元素 
				double tempSum = 0.0;
				for (int k = j; k <= i - 1; k++)
				{
					tempSum += resMat->m_Matrix[i][k] * resMat->m_Matrix[k][j];
				}
				resMat->m_Matrix[i][j] = -1.0*tempSum / resMat->m_Matrix[i][i];
			}

		}
	}

	return *resMat;

}

//矩阵求逆 上三角矩阵
Matrix& Matrix::InverseUpTriangle()
{
	//错误判断 方阵检测
	if (this->m_Row != this->m_Col)
	{
		std::cout << "Error: <InverseUpTriangle> Matrix Col != Row" << std::endl;
		return *this;
	}

	//上三角求逆
	Matrix* resMat = new Matrix(*this);

	for (int j = resMat->m_Col-1; j >=0; j--)
	{
		for (int i = j; i >=0; i--)
		{
			//分段求解 对角线为倒数
			if (i == j)
			{
				resMat->m_Matrix[i][j] = 1 / resMat->m_Matrix[i][j];
			}
			else
			{
				//分段求解 非对角线元素 
				double tempSum = 0.0;
				for (int k = j; k >= i+1; k--)
				{
					tempSum += resMat->m_Matrix[i][k] * resMat->m_Matrix[k][j];
				}
				resMat->m_Matrix[i][j] = -1.0 * tempSum / resMat->m_Matrix[i][i];
			}

		}
	}

	return *resMat;
}

//矩阵LU分解 顺序分解 对于病态矩阵可能存在精度问题
void Matrix::ResolveLU(Matrix& LMat, Matrix& UMat)
{
	if (this->m_Col != this->m_Row)
	{
		std::cout << "Error: <ResolveLU> Is Not Square Matrix" << std::endl;
		return;
	}

	//存在性判定 顺序主子式不为0
	for (int i = 0; i < this->m_Row; i++)
	{
		if (this->Det(i) == 0)
		{
			std::cout << "Error: <ResolveLU> order Det = 0" << std::endl;
			return;
		}
	}

	//LU 分解
	//L矩阵为单位矩阵
	LMat = this->Uint();

	//U矩阵初始化为空矩阵
	Matrix temp(this->m_Row, this->m_Col);
	UMat = temp;

	for (int i = 0; i < this->m_Row; i++)
	{
		//计算U
		for (int j1 = i; j1 < this->m_Col; j1++)
		{
			double tempSum1 = 0.0;

			if (i != 0)
			{
				for (int j2 = 0; j2 <= i - 1; j2++)
				{
					tempSum1 += LMat.m_Matrix[i][j2] * UMat.m_Matrix[j2][j1];
				}
			}

			UMat.m_Matrix[i][j1] = this->m_Matrix[i][j1] - tempSum1;

		}

		//计算L
		for (int i1 = i; i1 < this->m_Row; i1++)
		{
			double tempSum2 = 0.0;

			if (i != 0)
			{
				for (int j2 = 0; j2 <= i - 1; j2++)
				{
					tempSum2 += LMat.m_Matrix[i1][j2] * UMat.m_Matrix[j2][i];
				}
			}

			LMat.m_Matrix[i1][i] = (this->m_Matrix[i1][i] - tempSum2)/UMat.m_Matrix[i][i];

		}
	}

}

//矩阵的LUP分解 P*A = L*U 添加了列主消元功能 
//L为主对角线元素为1的下三角矩阵 U为上二角矩阵 P为行交换矩阵 P*A=L*U
void Matrix::ResolveLUP(Matrix& LMat, Matrix& UMat, Matrix& PMat)
{
	//条件判断 矩阵行列式不为0
	if (this->Det() == 0)
	{
		std::cout << "Error: <ResolveLUP> Can't Resolve Matrix To L U P" << std::endl;
		return;
	}

	//初始化 L U P
	LMat = this->Uint();
	PMat = this->Uint();
	UMat = *this;

	//进行分解计算
	for (int i = 0; i < UMat.m_Row - 1; i++)
	{
		//记录最大行所在行标
		int tempMaxRow = i;

		for (int i1 = i + 1; i1 < UMat.m_Row; i1++)
		{
			if (abs(UMat.m_Matrix[i1][i]) > abs(UMat.m_Matrix[tempMaxRow][i]))
			{
				tempMaxRow = i1;
			}
		}

		//进行交换 将当前第i行与第tempMaxRow行进行互换 初等行变换
		UMat = UMat.SwapRow(i, tempMaxRow);

		//L矩阵做出对应交换 先交换<itempMaxRow>列再交换<itempMaxRow>行
		LMat = LMat.SwapCol(i, tempMaxRow);
		LMat = LMat.SwapRow(i, tempMaxRow);

		//P矩阵做出对应变换 交换<itempMaxRow>行
		PMat = PMat.SwapRow(i, tempMaxRow);

		//高斯消元 V矩阵消除下三角区域，L矩阵添加下三角区域
		for (int i1 = i + 1; i1 < UMat.m_Row; i1++)
		{
			//记录消元系数
			double deleteVar = UMat.m_Matrix[i1][i] / UMat.m_Matrix[i][i];

			//L矩阵列填充
			LMat.m_Matrix[i1][i] = deleteVar;

			//U矩阵列消除
			UMat = UMat.MultRow(UMat.m_Matrix[i][i], i1).AddRow(i1, i, -1.0 * UMat.m_Matrix[i1][i]).MultRow(1 / UMat.m_Matrix[i][i], i1);
		}
	}

	return;
}