Eigen入門之密集矩陣 2-- Matrix及Vector的計算方法

簡介

Eigen內的Matrix和Vector提供了類似C++的運算符，如+，-,*；也提供了編程的函數方法，如點乘和叉乘的dot(), cross()，如此等等。

在Eigen的Matrix類，代表矩陣matrics和向量vector，重載的運算符僅用於支持線性代數的運算，而不支持標量計算。比如matrix1 * matrix2，表示矩陣matrix 乘以 matrix2，而matrix1 + 10則不允許。

加法和減法

如大家所知，如果2個矩陣運行運算，對2個矩陣的行數和列數是有條件要求的。另外，在Eigen內，用於計算時，矩陣的係數類型必須統一，並不會內部進行類型轉換。

二元運算符 + ： ma + mb;

二元運算符 - ： ma - mb;

一元運算符 - ： - ma;

組合運算符 +=： ma+=mb;

組合運算符 -=： ma-=mb;

示例：

// matrix_cal1.cpp
#include <iostream>
#include <Eigen/Dense>

using namespace Eigen;

int main()
{
  Matrix2d a;
  a << 1, 2,
       3, 4;

  MatrixXd b(2,2);
  b << 5, 6,
       7, 8;

  std::cout << "a + b =\n" << a + b << std::endl;
  std::cout << "a - b =\n" << a - b << std::endl;
  std::cout << "Doing a += b;" << std::endl;
  a += b;
  std::cout << "Now a =\n" << a << std::endl;

  std::cout << endl;
  Vector3d v(1,2,3);
  Vector3d w(1,0,0);
  std::cout << "-v + w - v =\n" << -v + w - v << std::endl;
}

執行後輸出：

$ ./matrix_cal1 
a + b =
 6  8
10 12
a - b =
-4 -4
-4 -4
Doing a += b;
Now a =
 6  8
10 12
-v + w - v =
-1
-4
-6

與標量的乘、除

與標量的乘法和除法計算是支持的，重載的操作符如下：

二元運算符 * ： matrix * scalar;

二元運算符 * ： scalar * matrix;

二元運算符 / ： matrix / scalar;

組合運算符 *=： matrix *= scalar;

組合運算符 /=： matrix /= scalar;

示例：

// matrix_cal2.cpp
#include <iostream>
#include <Eigen/Dense>

using namespace Eigen;

int main()
{
  Matrix2d a;
  a << 1, 2,
       3, 4;

  std::cout << "a * 2.5 =\n" << a * 2.5 << std::endl;

  Vector3d v(1,2,3);
  std::cout << "calculating v *= 2;" << std::endl;
  v *= 2;
  std::cout << "2.0 * v =\n" << 2.0 * v << std::endl;
  std::cout << "Now v =\n" << v << std::endl;

}

執行結果：

$ ./matrix_cal2
a * 2.5 =
2.5   5
7.5  10
calculating v *= 2;
2.0 * v =
 4
 8
12
Now v =
2
4
6

轉置和共軛

在線性代數中，矩陣有轉置操作，共軛計算，共軛轉置計算: $a^T$ , $\bar a$ , $a^*$ , Matrix提供了對應的函數：transpose(), conjugate(), and adjoint()。
說明一下：共軛是針對複數而言的，共軛矩陣也是複數矩陣的對應的共軛矩陣。

示例程序：

// matrix_cal3.cpp
#include <iostream>
#include <Eigen/Dense>

using namespace Eigen;
using namespace std;

int main()
{
    MatrixXcf a = MatrixXcf::Random(2,2);

    cout << "Here is the matrix a\n" << a << endl;
    cout << "Here is the matrix a^T\n" << a.transpose() << endl;
    cout << "Here is the conjugate of a\n" << a.conjugate() << endl;
    cout << "Here is the matrix a^*\n" << a.adjoint() << endl;
}

執行：

$ g++   matrix_cal3.cpp -o matrix_cal3 -I /usr/local/include/eigen3
$ ./matrix_cal3
Here is the matrix a
(-0.999984,-0.736924) (0.0655345,-0.562082)
(0.511211,-0.0826997)  (-0.905911,0.357729)
Here is the matrix a^T
(-0.999984,-0.736924) (0.511211,-0.0826997)
(0.0655345,-0.562082)  (-0.905911,0.357729)
Here is the conjugate of a
 (-0.999984,0.736924)  (0.0655345,0.562082)
 (0.511211,0.0826997) (-0.905911,-0.357729)
Here is the matrix a^*
 (-0.999984,0.736924)  (0.511211,0.0826997)
 (0.0655345,0.562082) (-0.905911,-0.357729)

對實數矩陣，並沒有對應的共軛矩陣，其共軛轉置矩陣就是其轉置矩陣。

matrix-matrix與matrix-vector的乘法

矩陣的matrix-matrix乘法，及矩陣*矩陣，是由操作符operator *完成的。而在Eigen內，向量vector也是一種特殊的矩陣，所以，matrix-vector和vector-vector的乘法也是由operator *來完成。

這裏有2種運算：

二元運算符 * ： a * b;
複合運算符 *=: a *= b;

示例：

// matrix_cal4.cpp
#include <iostream>
#include <Eigen/Dense>

using namespace Eigen;

int main()
{
  Matrix2d mat;
  mat << 1, 2,
         3, 4;

  Vector2d u(-1,1), v(2,0);

  std::cout << "Here is mat*mat:\n" << mat*mat << std::endl;
  std::cout << "Here is mat*u:\n" << mat*u << std::endl;
  std::cout << "Here is u^T*mat:\n" << u.transpose()*mat << std::endl;
  std::cout << "Here is u^T*v:\n" << u.transpose()*v << std::endl;
  std::cout << "Here is u*v^T:\n" << u*v.transpose() << std::endl;
  std::cout << "Let's multiply mat by itself" << std::endl;

  mat = mat*mat;

  std::cout << "Now mat is mat:\n" << mat << std::endl;
}

執行

$ g++   matrix_cal4.cpp -o matrix_cal4 -I /usr/local/include/eigen3
$
$ ./matrix_cal4
Here is mat*mat:
 7 10
15 22
Here is mat*u:
1
1
Here is u^T*mat:
2 2
Here is u^T*v:
-2
Here is u*v^T:
-2 -0
 2  0
Let's multiply mat by itself
Now mat is mat:
 7 10
15 22

aliasing issues與運算模板

矩陣的點乘與叉乘

使用matrix類的dot(), cross()函數，來執行矩陣的點乘和叉乘。

點乘的結果可以看出是1X1的矩陣，u * v 還可以使用u.adjoint()*v進行計算.

示例：

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

using namespace Eigen;
using namespace std;

int main()
{
  Vector3d v(1,2,3);
  Vector3d w(0,1,2);
  cout << "Dot product: " << v.dot(w) << endl;
  double dp = v.adjoint() * w; // automatic conversion of the inner product to a scalar
  cout << "Dot product via a matrix product: " << dp << endl;
  cout << "Cross product:\n" << v.cross(w) << endl;
}

執行輸出：

Dot product: 8
Dot product via a matrix product: 8
Cross product:
1
-2
1

基礎算數操作–arithmetic reduction

對矩陣matrix或向量vector，提供了一些算數分解操作，比如獲取矩陣的係數之和，最大值，最小值，平均值、及對角線的相關算數。

sum() : 係數之和
prod() : 係數乘積
maxCoeff() : 最大系數
minCoeff() : 最小系數
trace() : 對角線係數和。

#include <iostream>
#include <Eigen/Dense>
using namespace std;
int main()
{
  Eigen::Matrix2d mat;
  mat << 1, 2,
         3, 4;
  cout << "Here is mat.sum():       " << mat.sum()       << endl;   // 10
  cout << "Here is mat.prod():      " << mat.prod()      << endl;   // 24
  cout << "Here is mat.mean():      " << mat.mean()      << endl;   // 2.5
  cout << "Here is mat.minCoeff():  " << mat.minCoeff()  << endl;   // 1
  cout << "Here is mat.maxCoeff():  " << mat.maxCoeff()  << endl;   // 4 
  cout << "Here is mat.trace():     " << mat.trace()     << endl;   // 5
}

注意這些函數的重載,還可以同時獲取該係數的位置。

// matrix_cal5.cpp
#include <iostream>
#include <Eigen/Dense>

using namespace Eigen;
using namespace std;

int main()
{
  Matrix3f m = Matrix3f::Random();
  std::ptrdiff_t i, j;

  float minOfM = m.minCoeff(&i,&j);

  cout << "Here is the matrix m:\n" << m << endl;
  cout << "Its minimum coefficient (" << minOfM 
       << ") is at position (" << i << "," << j << ")\n\n";

  RowVector4i v = RowVector4i::Random();
  int maxOfV = v.maxCoeff(&i);

  cout << "Here is the vector v: " << v << endl;
  cout << "Its maximum coefficient (" << maxOfV 
       << ") is at position " << i << endl;

}

執行結果如下：

$ g++   matrix_cal5.cpp -o matrix_cal5 -I /usr/local/include/eigen3
$ ./matrix_cal5
Here is the matrix m:
 -0.999984 -0.0826997  -0.905911
 -0.736924  0.0655345   0.357729
  0.511211  -0.562082   0.358593
Its minimum coefficient (-0.999984) is at position (0,0)

Here is the vector v:  933495885 -250177384   41696341  710742668
Its maximum coefficient (933495885) is at position 0

Eigen入門之密集矩陣 2-- Matrix及Vector的計算方法

簡介

加法和減法

與標量的乘、除

轉置和共軛

matrix-matrix與matrix-vector的乘法

aliasing issues與運算模板

矩陣的點乘與叉乘

基礎算數操作–arithmetic reduction

Git提交branch到remote

Eigen入門之密集矩陣 9 - 別名混亂Aliasing

NLog配置文件介紹

Eigen 入門 1 - Eigen簡介

安裝cygwin時的一個長時間處理

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