C++使用eigen库做本征分解(eigendecomposition)

Eigendecomposition的概念可见https://en.wikipedia.org/wiki/Eigendecomposition_of_a_matrix

这里贴一段厄米矩阵的代码，见https://eigen.tuxfamily.org/dox/group__TutorialLinearAlgebra.html

注意，不同本征值的本征向量是正交的，这是我们可以直接用矩阵共轭来取代矩阵求逆的原因。

 1 #include <iostream>
 2 #include <eigen3/Eigen/Dense>
 3 using namespace std;
 4 using namespace Eigen;
 5 
 6 int main ()
 7 {
 8   Matrix2cd A;
 9   A<<complex<double>(1,0), complex<double>(0,1),
10     complex<double>(0,-1), complex<double>(1,0);
11 
12   SelfAdjointEigenSolver<Matrix2cd> solver(A);
13   if (solver.info() != Success)
14     {
15       cerr<<"Eigen solver failed."<<endl;
16       abort ();
17     }
18   Matrix2cd lambda = Matrix2cd::Zero();
19   for (int i = 0; i < lambda.cols(); ++i)
20     lambda(i,i) = solver.eigenvalues()(i);
21   Matrix2cd Q = solver.eigenvectors();
22   cout<<"Matrix A:\n"<<A<<endl<<endl;
23   cout<<"Matrix lambda:\n"<<lambda<<endl<<endl;
24   cout<<"Matrix Q:\n"<<Q<<endl<<endl;
25   cout<<"Q*Q^dagger:\n"<<Q*Q.adjoint()<<endl<<endl;
26   cout<<"Q*lambda*Q^dagger:\n"<<Q*lambda*Q.adjoint()<<endl<<endl;
27 
28   return 0;
29 }

输出结果为

 1 Matrix A:
 2  (1,0)  (0,1)
 3 (0,-1)  (1,0)
 4 
 5 Matrix lambda:
 6 (0,0) (0,0)
 7 (0,0) (2,0)
 8 
 9 Matrix Q:
10  (0.707107,0)  (0.707107,0)
11  (0,0.707107) (0,-0.707107)
12 
13 Q*Q^dagger:
14 (1,0) (0,0)
15 (0,0) (1,0)
16 
17 Q*lambda*Q^dagger:
18  (1,0)  (0,1)
19 (0,-1)  (1,0)