运算

+,- 加减法

• 逐元加减法

#include <iostream>
#include "e:/eigen/Eigen/Dense"
using namespace std;

int main()
{
  Eigen::Matrix2d a;
  a << 1, 2,
       3, 4;
  Eigen::MatrixXd b(2,2);
  b << 10, 20,
       30, 40;
  cout << "a + b =\n" << a + b << endl;
  cout << "a - b =\n" << a - b << endl;
  cout << "Doing a += b;" << endl;
  a += b;
  cout << "Now a =\n" << a << endl;
  Eigen::Vector3d v(1,2,3);
  Eigen::Vector3d w(1,0,0);
  cout << "-v + w - v =\n" << -v + w - v << endl;
}

a + b =
11 22
33 44
a - b =
 -9 -18
-27 -36
Doing a += b;
Now a =
11 22
33 44
-v + w - v =
-1
-4
-6

Process returned 0 (0x0)   execution time : 0.573 s
Press any key to continue.

* / 乘除法

逐元 乘法

1、matrix*scalar
2、scalar*matrix
3、matrix*=scalar

#include <iostream>
#include <Eigen/Dense>

int main()
{
  Eigen::Matrix2d a;
  a << 10, 20,
       30, 40;
  Eigen::Vector3d v(1,2,3);
  std::cout << "a * 0.1 =\n" << a * 0.1 << std::endl;
  std::cout << "0.1 * v =\n" << 10 * v << std::endl;
  std::cout << "Doing v *= 2;" << std::endl;
  v *= 2;
  std::cout << "Now v =\n" << v << std::endl;
}

a * 2.5 =
1 2
3 4
0.1 * v =
10
20
30
Doing v *= 2;
Now v =
2
4
6

Process returned 0 (0x0)   execution time : 0.534 s
Press any key to continue.

逐元 除法

    1、matrix/scalar
   2、matrix/=scalar

#include <iostream>
#include "e:/eigen/Eigen/Dense"
using namespace std;
int main()
{
  Eigen::Matrix2d a;
  a << 10, 20,
       30, 40;
  Eigen::Vector3f v(1,2,3);
  cout << "a / 5 =\n" << a / 5 << endl;
  cout << "v / 5 =\n" << v /5 << endl;
  cout << "Doing a /= 10;" << endl;
  a /= 10;
  cout << "Now a =\n" << a << endl;
}

a / 5 =
2 4
6 8
v / 5 =
0.2
0.4
0.6
Doing a /= 10;
Now a =
1 2
3 4

Process returned 0 (0x0)   execution time : 0.477 s
Press any key to continue.

逐元综合运算

#include <iostream>
#include "e:/eigen/Eigen/Dense"
using namespace std;
int main()
{
  Eigen::Matrix2d a;
  a << 10, 20,
       30, 40;
  Eigen::Matrix2d b;
  b << 1, 2,
       3, 4;
  cout << "a *10+ b =\n" << a*10+b << endl;
}

a *10+ b =
101 202
303 404

Process returned 0 (0x0)   execution time : 0.394 s
Press any key to continue.

矩阵乘法与加减法

#include <iostream>
#include "e:/eigen/Eigen/Dense"
using namespace std;
int main()
{
  Eigen::Matrix2d a;
  a << 10, 20,
       30, 40;
  Eigen::Matrix2d b;
  b << 2, 4,
       8, 16;
  Eigen::Vector2d v(1,2);
  cout << "a * b =\n" << a *b << endl;
  cout << "a * v =\n" << a *v << endl;
  cout << "a + b =\n" << a +b << endl;
  cout << "a - b =\n" << a -b << endl;
}

a * b =
180 360
380 760
a * v =
 50
110
a + b =
12 24
38 56
a - b =
 8 16
22 24

Process returned 0 (0x0)   execution time : 0.429 s
Press any key to continue.

转置、共轭、伴随矩阵

• 共轭矩阵相关知识

• 转置矩阵
  将矩阵的行列互换得到的新矩阵称为转置矩阵，转置矩阵的行列式不变。
  

• 复数:

template <class Type>
class complex

引用自https://learn.microsoft.com/zh-cn/cpp/standard-library/complex-class?view=msvc-170
复数 a + bi
名称 描述
imag 提取复数的虚分量。
real 提取复数的实分量。

#include <complex>
#include <iostream>

int main( ) {
    using namespace std;

    complex<double> c1( 4.0 , 3.0 );
    cout << "The complex number c1 = " << c1 << endl;

    double dr1 = c1.real();
    cout << "The real part of c1 is c1.real() = "
        << dr1 << "." << endl;

    double di1 = c1.imag();
    cout << "The imaginary part of c1 is c1.imag() = "
        << di1 << "." << endl; } 

 The complex number c1 = (4,3)
The real part of c1 is c1.real() = 4.
The imaginary part of c1 is c1.imag() = 3.

• 复数矩阵
  typedef Matrix< std::complex< float >, Dynamic, Dynamic > Eigen::MatrixXcf
• example

#include <iostream>
#include "e:/eigen/Eigen/Dense"
using namespace std;
int main()
{
  Eigen::Matrix2d a;
  a << 10, 20,
       30, 40;
  Eigen::MatrixXcf b= Eigen::MatrixXcf::Random(2,2);
  cout << "转置：a^T =\n" << a.transpose() << endl;
  cout << "共轭：a conjugate() =\n" << b.conjugate() << endl;
  cout << "伴随矩阵：a adjoint() =\n"<< a.adjoint() << endl;
}

转置：a^T =
10 30
20 40
共轭：a conjugate() =
   (0.127171,0.997497) (-0.0402539,-0.170019)
   (0.617481,0.613392)    (0.791925,0.299417)
伴随矩阵：a adjoint() =
10 30
20 40

Process returned 0 (0x0)   execution time : 0.540 s
Press any key to continue.

• a = a.transpose()转置并替换使用a.transposeInPlace()
• a = a.adjoint() 共轭并替换a.adjointInPlace()

点乘法,叉积

• dot product

引用自百度百科的解释
点积在数学中，又称数量积（dot product; scalar product），是指接受在实数R上的两个向量并返回一个实数值标量的二元运算。它是欧几里得空间的标准内积。
两个向量a = [a1, a2,…, an]和b = [b1, b2,…, bn]的点积定义为：
a·b=a1b1+a2b2+……+anbn。
使用矩阵乘法并把（纵列）向量当作n×1 矩阵，点积还可以写为：
$a\cdot b=a^T\ast b，这里的a^T指示矩阵a的转置。$

• cross product

引用自百度百科知识
向量积，数学中又称外积、叉积，物理中称矢积、叉乘，是一种在向量空间中向量的二元运算。与点积不同，它的运算结果是一个向量而不是一个标量。并且两个向量的叉积与这两个向量和垂直。其应用也十分广泛，通常应用于物理学光学和计算机图形学中。

• example

#include <iostream>
#include "e:/eigen/Eigen/Dense"
using namespace std;
int main()
{
  Eigen::Vector3d v(1,2,3);
  Eigen::Vector3d w(0,1,2);
  cout << "v dot product w =\n" << v.dot(w) << endl;
  cout << "v cross product w =\n" << v.cross(w) << endl;
}

v dot product w =
8
v cross product w =
 1
-2
 1

Process returned 0 (0x0)   execution time : 0.327 s
Press any key to continue.