How to change property of Eigen vectors

Question

I calculated the Eigen values & vectors of two $K$ and $M$ matrices with LAPACK DGGEVX routine. I need to make Eigen vectors orthonormal like this: $\Phi^T M \Phi = I$ . How can I do that?

Answer 1

If you have a generalized eigenproblem $Kx=\lambda M x$ where $K$ is symmetric and $M$ is positive definite, then the generalized eigenvectors are $M$-orthogonal. This means that if $\Phi$ is the eigenvector matrix then $\Phi^T M \Phi$ is diagonal with positive diagonal entries. It also typically means that they are not Euclidean-orthogonal, that is, that $\Phi^T \Phi$ is not diagonal.

To make the eigenvectors $M$-orthonormal, one can divide each eigenvector by the square root of the corresponding diagonal entry. Then you have $\Phi^T M \Phi = I$.