Eigen库求解特征值特征向量

71 阅读 0 评论 47 点赞

我是靠谱客的博主务实寒风，最近开发中收集的这篇文章主要介绍Eigen库求解特征值特征向量，觉得挺不错的，现在分享给大家，希望可以做个参考。

概述

http://eigen.tuxfamily.org/dox/classEigen_1_1EigenSolver.html

Eigen中的矩阵类型一般都是用类似MatrixXXX来表示，可以根据该名字来判断其数据类型，比如说’d’代表double并不是用来表示整数的,;‘f’代表float; ‘i’代表整数;‘c’代表complex，即复数；’d’表示dynamic，即表示矩阵中有些维数是不确定的，动态的……举例子比如说：Matrix2cd，表示的是2*2维的，其每个元素都是复数，复数的实部和虚部都为double类型。

EigenSolver	(	const MatrixType &	matrix,
		bool	computeEigenvectors = `true`
	)

inline

Constructor; computes eigendecomposition of given matrix.

Parameters

[in]	matrix	Square matrix whose eigendecomposition is to be computed.
[in]	computeEigenvectors	If true, both the eigenvectors and the eigenvalues are computed; if false, only the eigenvalues are computed.

This constructor calls compute() to compute the eigenvalues and eigenvectors.

Example:

     MatrixXd A = 
     MatrixXd::Random(6,6); 
   
     cout << 
     "Here is a random 6x6 matrix, A:" << endl << A << endl << endl; 
   
     EigenSolver<MatrixXd> es(A); 
   
     cout << 
     "The eigenvalues of A are:" << endl << es.eigenvalues() << endl; 
   
     cout << 
     "The matrix of eigenvectors, V, is:" << endl << es.eigenvectors() << endl << endl; 
   
     complex<double> lambda = es.eigenvalues()[0]; 
   
     cout << 
     "Consider the first eigenvalue, lambda = " << lambda << endl; 
   
    VectorXcd v = es.eigenvectors().col(0); 
   
     cout << 
     "If v is the corresponding eigenvector, then lambda * v = " << endl << lambda * v << endl; 
   
     cout << 
     "... and A * v = " << endl << A.cast<complex<double> >() * v << endl << endl; 
   
    MatrixXcd D = es.eigenvalues().asDiagonal(); 
   
    MatrixXcd V = es.eigenvectors(); 
   
     cout << 
     "Finally, V * D * V^(-1) = " << endl << V * D * V.inverse() << endl;

Output:

Here is a random 6x6 matrix, A:
   0.68   -0.33   -0.27  -0.717  -0.687  0.0259
 -0.211   0.536  0.0268   0.214  -0.198   0.678
  0.566  -0.444   0.904  -0.967   -0.74   0.225
  0.597   0.108   0.832  -0.514  -0.782  -0.408
  0.823 -0.0452   0.271  -0.726   0.998   0.275
 -0.605   0.258   0.435   0.608  -0.563  0.0486

The eigenvalues of A are:
  (0.049,1.06)
 (0.049,-1.06)
     (0.967,0)
     (0.353,0)
 (0.618,0.129)
(0.618,-0.129)
The matrix of eigenvectors, V, is:
 (-0.292,-0.454)   (-0.292,0.454)      (-0.0607,0)       (-0.733,0)    (0.59,-0.122)     (0.59,0.122)
  (0.134,-0.104)    (0.134,0.104)       (-0.799,0)        (0.136,0)    (0.335,0.368)   (0.335,-0.368)
  (-0.422,-0.18)    (-0.422,0.18)        (0.192,0)       (0.0563,0)  (-0.335,-0.143)   (-0.335,0.143)
 (-0.589,0.0274) (-0.589,-0.0274)      (-0.0788,0)       (-0.627,0)   (0.322,-0.156)    (0.322,0.156)
  (-0.248,0.132)  (-0.248,-0.132)        (0.401,0)        (0.218,0)  (-0.335,-0.076)   (-0.335,0.076)
    (0.105,0.18)    (0.105,-0.18)       (-0.392,0)     (-0.00564,0)  (-0.0324,0.103) (-0.0324,-0.103)

Consider the first eigenvalue, lambda = (0.049,1.06)
If v is the corresponding eigenvector, then lambda * v = 
  (0.466,-0.331)
   (0.117,0.137)
   (0.17,-0.456)
(-0.0578,-0.622)
 (-0.152,-0.256)
   (-0.186,0.12)
... and A * v = 
  (0.466,-0.331)
   (0.117,0.137)
   (0.17,-0.456)
(-0.0578,-0.622)
 (-0.152,-0.256)
   (-0.186,0.12)

Finally, V * D * V^(-1) = 
     (0.68,1.9e-16)    (-0.33,4.82e-17)   (-0.27,-2.37e-16)    (-0.717,1.6e-16)   (-0.687,-2.2e-16)   (0.0259,2.72e-16)
  (-0.211,2.22e-16)    (0.536,4.16e-17)  (0.0268,-2.98e-16)           (0.214,0)   (-0.198,6.66e-16)    (0.678,6.66e-16)
   (0.566,1.22e-15)   (-0.444,1.11e-16)   (0.904,-4.61e-16)  (-0.967,-3.61e-16)    (-0.74,7.22e-16)    (0.225,8.88e-16)
    (0.597,1.6e-15)    (0.108,1.84e-16)    (0.832,-5.6e-16)  (-0.514,-4.44e-16)   (-0.782,1.28e-15)   (-0.408,9.44e-16)
  (0.823,-8.33e-16) (-0.0452,-2.71e-16)    (0.271,5.53e-16)   (-0.726,7.77e-16)   (0.998,-2.33e-15)   (0.275,-1.67e-15)
  (-0.605,1.03e-15)    (0.258,1.91e-16)    (0.435,-4.6e-16)   (0.608,-6.38e-16)   (-0.563,1.69e-15)   (0.0486,1.25e-15)