[Math] Generalized eigenvalue problem with symmetric positive semidefinite matrix

Question

$SV = TVD$ where $S$ is symmetric positive semi definite, $T$ is symmetric positive definite, $V$ is eigenvector matrix and $D$ is eigenvalue matrix.

In this case, can we guarantee $V$ is orthogonal? i.e. $V^{H} * V = $I?

Answer 1

I think this is not true in general. You can however prove, that the eigenvectors are $T$- and $S$-orthogonal:

Let $v_i$, $v_j$ be eigenvectors to eigenvalues $\lambda_i\ne\lambda_j$. Then we have $$ v_i^HSv_j = \lambda_j v_i^HTv_j, \quad v_j^HSv_i = \lambda_i v_j^HTv_i. $$ Hence it holds, $$ \lambda_j v_i^HTv_j = \overline{\lambda_i v_j^HTv_i} = \lambda_i v_i^HTv_j, $$ which implies $$ v_i^HTv_j = 0. $$ If both $\lambda_i\ne0$, $\lambda_j\ne0$ then also $$ v_i^HSv_j = 0 $$ can be proven (with analogous arguments).