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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!

Uri Cohen
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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.

Ben Grossmann
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  • Thanks. For the general rectangular $A$ case you just state that the eigenvalues are real, right? I'm hoping they can be related to $d_i$ and $\sigma_i$, the diagonal elements of $D$ and $\Sigma$. – Uri Cohen Apr 03 '14 at 14:42
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    Not really. For example, if each $d_i > 0$, then for any symmetric matrix $B$: there exists an $A$ such that $ADA^T = B$. – Ben Grossmann Apr 03 '14 at 17:42