Eigenvalues of $ADA^T$

Question

Consider a rectangular matrix $A\in\mathbb{R}^{M\times N}$ and a diagonal matrix $D\in\mathbb{R}^{N\times N}$. What can one say on the eigenvalues and eigenvectors of $ADA^T$?

For example, if we denote $\{d_i\}_{i=1..N}$ the diagonal components of $D$ and $A=U\Sigma V^T$ is its singular values decompositions, can you express the eigenvalues and eigenvectors of $ADA^T$ in some simple way using $D,U,\Sigma,V$?

Thanks!

Answer 1

Note that $(ADA^T)^T = ADA^T$. Since $ADA^T$ is symmetric, its eigenvalues are real and we may select an orthonormal basis of eigenvectors by the spectral theorem.

Besides that, note that if $A$ is square, the product is congruent to $D$. If $A$ also has full rank, then Sylvester's law of inertia applies.