c ++特征值和对应于最小特征值的特征向量

Question

I am trying to find out the eigenvalues and the eigenvector corresponding to the smallest eigenvalue. I have a matrix A (nx2) and I have computed B = transpose(A) * a . When I am using c++ eigen function compute() and print the eigenvalues of matrix B , it shows something like this:

(4.4, 0)
(72.1, 0)

Printing the eigenvectors it gives output:

(-0.97, 0)   (0.209, 0)
(-0.209, 0)  (-0.97, 0)

I am confused. Eigenvectors can't be zero I guess. So, for the smallest eigenvalue 4.4, is the corresponding eigenvector (-0.97, -0.209) ?

PS - when I print

mysolution.eigenvalues()[0]

it prints (4.4, 0) . And when I print

mysolution.eigenvectors().col(0)

it prints (-0.97, 0) (0.209, 0) . That's why I guess I can assume that for eigenvalue 4.4, the corresponding eigenvector is (-0.97, -0.209) .

Answer 1

I guess you are correct.

None of your eigenvalues is null, though. It seems that you are working with complex numbers.

Could it be that you selected a complex floating point matrix to do your computations? Something along the lines of MatrixX2cf or MatrixX2cd .

Answer 2

Every square matrix has a set of eigenvalues. But even if the matrix itself only consists of real numbers, the eigenvalues and -vectors might contain complex numbers (take (0 1;-1 0) for example)

If Eigen knows nothing about your matrix structure (ie is it symmetric/self-adjoint? Is it orthonormal/unitary?) but still wants to provide you with exact eigenvalues, the only general type that can hold all possible eigenvalues is a complex number.

Thus, Eigen always returns complex numbers which are represented as pairs (a, b) for a + bi. Eigen will only return real numbers if the matrix is self-adjoint, ie SelfAdjointView is used to access the matrix.

If you know for a fact that your matrix only has real eigenvalues, you can just extract the real part by eigenvalue.real since Eigen returns std::complex values.

EDIT: I just realized that if your matrix A has no complex entries, B=transposed(A)*A is self-adjoint and thus you could just use a SelfAdjointView of the matrix to compute the real eigenvalues and -vectors.