Eigen入门之密集矩阵 2-- Matrix及Vector的计算方法

简介

Eigen内的Matrix和Vector提供了类似C++的运算符，如+，-,*；也提供了编程的函数方法，如点乘和叉乘的dot(), cross()，如此等等。

在Eigen的Matrix类，代表矩阵matrics和向量vector，重载的运算符仅用于支持线性代数的运算，而不支持标量计算。比如matrix1 * matrix2，表示矩阵matrix 乘以 matrix2，而matrix1 + 10则不允许。

加法和减法

如大家所知，如果2个矩阵运行运算，对2个矩阵的行数和列数是有条件要求的。另外，在Eigen内，用于计算时，矩阵的系数类型必须统一，并不会内部进行类型转换。

二元运算符 + ： ma + mb;

二元运算符 - ： ma - mb;

一元运算符 - ： - ma;

组合运算符 +=： ma+=mb;

组合运算符 -=： 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;
}

执行后输出：

$ ./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

与标量的乘、除

与标量的乘法和除法计算是支持的，重载的操作符如下：

二元运算符 * ： matrix * scalar;

二元运算符 * ： scalar * matrix;

二元运算符 / ： matrix / scalar;

组合运算符 *=： matrix *= scalar;

组合运算符 /=： 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;}

执行结果：

$ ./matrix_cal2
a * 2.5 =
2.5   5
7.5  10
calculating v *= 2;
2.0 * v =48
12
Now v =
2
4
6

转置和共轭

在线性代数中，矩阵有转置操作，共轭计算，共轭转置计算: aTa^TaT, aˉ\bar aaˉ, a∗a^*a∗, Matrix提供了对应的函数：transpose(), conjugate(), and adjoint()。
说明一下：共轭是针对复数而言的，共轭矩阵也是复数矩阵的对应的共轭矩阵。

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;
}

执行：

$ 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)

对实数矩阵，并没有对应的共轭矩阵，其共轭转置矩阵就是其转置矩阵。

matrix-matrix与matrix-vector的乘法

矩阵的matrix-matrix乘法，及矩阵*矩阵，是由操作符operator *完成的。而在Eigen内，向量vector也是一种特殊的矩阵，所以，matrix-vector和vector-vector的乘法也是由operator *来完成。

这里有2种运算：

二元运算符 * ： a * b;
复合运算符 *=: 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;
}

执行

$ 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 -02  0
Let's multiply mat by itself
Now mat is mat:7 10
15 22

aliasing issues与运算模板

矩阵的点乘与叉乘

使用matrix类的dot(), cross()函数，来执行矩阵的点乘和叉乘。

点乘的结果可以看出是1X1的矩阵，u * v 还可以使用u.adjoint()*v进行计算.

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 scalarcout << "Dot product via a matrix product: " << dp << endl;cout << "Cross product:\n" << v.cross(w) << endl;
}

执行输出：

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

基础算数操作–arithmetic reduction

对矩阵matrix或向量vector，提供了一些算数分解操作，比如获取矩阵的系数之和，最大值，最小值，平均值、及对角线的相关算数。

sum() : 系数之和
prod() : 系数乘积
maxCoeff() : 最大系数
minCoeff() : 最小系数
trace() : 对角线系数和。

#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;   // 10cout << "Here is mat.prod():      " << mat.prod()      << endl;   // 24cout << "Here is mat.mean():      " << mat.mean()      << endl;   // 2.5cout << "Here is mat.minCoeff():  " << mat.minCoeff()  << endl;   // 1cout << "Here is mat.maxCoeff():  " << mat.maxCoeff()  << endl;   // 4 cout << "Here is mat.trace():     " << mat.trace()     << endl;   // 5
}

注意这些函数的重载,还可以同时获取该系数的位置。

// 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;}

执行结果如下：

$ 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   0.3577290.511211  -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