Eigen 矩阵基本运算

矩阵和向量的运算
提供一些概述和细节：关于矩阵、向量以及标量的运算。

1. 介绍

Eigen提供了matrix/vector的运算操作，既包括重载了c++的算术运算符+/-/*，也引入了一些特殊的运算比如点乘dot、叉乘cross等。

对于Matrix类（matrix和vectors）这些操作只支持线性代数运算，比如：matrix1*matrix2表示矩阵的乘机，vetor+scalar是不允许的。如果你想执行非线性代数操作，请看下一篇（暂时放下）。

2. 加减

左右两侧变量具有相同的尺寸（行和列），并且元素类型相同（Eigen不自动转化类型）操作包括：

二元运算 + 如a+b
二元运算 - 如a-b
一元运算 - 如-a
复合运算 += 如a+=b
复合运算 -= 如a-=b

#include <iostream>
 
#include <Eigen/Dense>
 
using namespace Eigen;
 
int main()
 
{
 
  Matrix2d a;
 
  a << 1, 2,
 
       3, 4;
 
  MatrixXd b(2,2);
 
  b << 2, 3,
 
       1, 4;
 
  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;
 
  Vector3d v(1,2,3);
 
  Vector3d w(1,0,0);
 
  std::cout << "-v + w - v =\n" << -v + w - v << std::endl;
 
}

输出：

a + b =

3 5

4 8

a - b =

-1 -1

2 0

Doing a += b;

Now a =

3 5

4 8

-v + w - v =

-1

-4

-6

3. 标量乘法和除法

乘/除标量是非常简单的，如下：

二元运算 * 如matrix*scalar
二元运算 * 如scalar*matrix
二元运算 / 如matrix/scalar
复合运算 *= 如matrix*=scalar
复合运算 /= 如matrix/=scalar

#include <iostream>
 
#include <Eigen/Dense>
 
using namespace Eigen;
 
int main()
 
{
 
  Matrix2d a;
 
  a << 1, 2,
 
       3, 4;
 
  Vector3d v(1,2,3);
 
  std::cout << "a * 2.5 =\n" << a * 2.5 << std::endl;
 
  std::cout << "0.1 * v =\n" << 0.1 * v << std::endl;
 
  std::cout << "Doing v *= 2;" << std::endl;
 
  v *= 2;
 
  std::cout << "Now v =\n" << v << std::endl;
 
}

结果

a * 2.5 =

2.5 5

7.5 10

0.1 * v =

0.1

0.2

0.3

Doing v *= 2;

Now v =

2

4

6

4. 表达式模板

这里简单介绍，在高级主题中会详细解释。在Eigen中，线性运算比如+不会对变量自身做任何操作，会返回一个“表达式对象”来描述被执行的计算。当整个表达式被评估完（一般是遇到=号），实际的操作才执行。

这样做主要是为了优化，比如

VectorXf a(50), b(50), c(50), d(50);

...

a = 3*b + 4*c + 5*d;

Eigen会编译这段代码最终遍历一次即可运算完成。

for(int i = 0; i < 50; ++i)

a[i] = 3*b[i] + 4*c[i] + 5*d[i];

因此，我们不必要担心大的线性表达式的运算效率。

5. 转置和共轭

表示transpose转置

表示conjugate共轭

表示adjoint(共轭转置) 伴随矩阵

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;

输出

Here is the matrix a

(-0.211,0.68) (-0.605,0.823)

(0.597,0.566) (0.536,-0.33)

Here is the matrix a^T

(-0.211,0.68) (0.597,0.566)

(-0.605,0.823) (0.536,-0.33)

Here is the conjugate of a

(-0.211,-0.68) (-0.605,-0.823)

(0.597,-0.566) (0.536,0.33)

Here is the matrix a^*

(-0.211,-0.68) (0.597,-0.566)

(-0.605,-0.823) (0.536,0.33)

对于实数矩阵，conjugate不执行任何操作，adjoint等价于transpose。

transpose和adjoint会简单的返回一个代理对象并不对本省做转置。如果执行 b=a.transpose() ，a不变，转置结果被赋值给b。如果执行 a=a.transpose() Eigen在转置结束之前结果会开始写入a，所以a的最终结果不一定等于a的转置。

Matrix2i a; a << 1, 2, 3, 4;
 
cout << "Here is the matrix a:\n" << a << endl;
 
a = a.transpose(); // !!! do NOT do this !!!
 
cout << "and the result of the aliasing effect:\n" << a << endl;

Here is the initial matrix a:

1 2 3

4 5 6

and after being transposed:

1 4

2 5

3 6

6. 矩阵-矩阵的乘法和矩阵-向量的乘法

向量也是一种矩阵，实质都是矩阵-矩阵的乘法。

二元运算 *如a*b
复合运算 *=如a*=b

#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;
 
}

输出

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

m=m*m并不会导致别名问题，Eigen在这里做了特殊处理，引入了临时变量。实质将编译为：

tmp = m*m

m = tmp

如果你确定矩阵乘法是安全的（并没有别名问题），你可以使用noalias()函数来避免临时变量 c.noalias() += a*b 。

7. 点运算和叉运算

dot()执行点积，cross()执行叉积，点运算得到1*1的矩阵。当然，点运算也可以用u.adjoint()*v来代替。

#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;
 
}

输出

Dot product: 8

Dot product via a matrix product: 8

Cross product:

1

-2

1

注意：点积只对三维vector有效。对于复数，Eigen的点积是第一个变量共轭和第二个变量的线性积。

8. 基础的归约操作

Eigen提供了而一些归约函数：sum()、prod()、maxCoeff()和minCoeff()，他们对所有元素进行操作。

#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;
 
  cout << "Here is mat.prod():      " << mat.prod()      << endl;
 
  cout << "Here is mat.mean():      " << mat.mean()      << endl;
 
  cout << "Here is mat.minCoeff():  " << mat.minCoeff()  << endl;
 
  cout << "Here is mat.maxCoeff():  " << mat.maxCoeff()  << endl;
 
  cout << "Here is mat.trace():     " << mat.trace()     << endl;
 
}

输出

Here is mat.sum(): 10

Here is mat.prod(): 24

Here is mat.mean(): 2.5

Here is mat.minCoeff(): 1

Here is mat.maxCoeff(): 4

Here is mat.trace(): 5

trace表示矩阵的迹，对角元素的和等价于 a.diagonal().sum() 。

minCoeff和maxCoeff函数也可以返回结果元素的位置信息。

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;

输出

Here is the matrix m:

0.68 0.597 -0.33

-0.211 0.823 0.536

0.566 -0.605 -0.444

Its minimum coefficient (-0.605) is at position (2,1)

 

Here is the vector v: 1 0 3 -3

Its maximum coefficient (3) is at position 2

9. 操作的有效性

Eigen会检测执行操作的有效性，在编译阶段Eigen会检测它们，错误信息是繁冗的，但错误信息会大写字母突出，比如:

Matrix3f m;
 
Vector4f v;
 
v = m*v;      // Compile-time error: YOU_MIXED_MATRICES_OF_DIFFERENT_SIZES

当然动态尺寸的错误要在运行时发现，如果在debug模式，assertions会触发后，程序将崩溃。

MatrixXf m(3,3);
 
VectorXf v(4);
 
v = m * v; // Run-time assertion failure here: "invalid matrix product"