I'm calculating eigenvectors of a high dimension complex symmetric matrix and I know this is the case about the Hermitian matrices but are eigenvectors of a complex symmetric matrix orthogonal too?
Because my matrix always has degenerate eigenvalues and the eigenvectors corresponding to those eigenvectors are not orthogonal. (Done with Matlab)
If not, can they form an orthogonal eigenbasis by orthogonalization?
What are the methods to do such thing? Will Gram-Schmidt be useful here?
EDIT: This matrix is a normal matrix.
Best Answer
Eigenvectors of a complex symmetric matrix are orthogonal in a tranpsose sense and not in the usual conjugate transpose sense. That is, if $\lambda_{1}$ (eigenvector $v_1$) and $\lambda_{2}$ (eigenvector $v_2$) are 2 distinct eigenvalues of a complex symmetric matrix A, then $v_1^T v_2=0$, but $v_1^\dagger v_2\neq 0$. eig() function in MATLAB will yield in such orthogonal eigenvectors in the transpose sense. You can check this for yourself in MATLAB. Hope I answered your question.