Bound on eigenvalues of matrices of the form $XDX^T$

Asked Oct 10 '18 at 23:35

Active Oct 10 '18 at 23:35

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I encountered this form of a Matrix while analyzing Logistic Regression (It's the Hessian).

Let $H = XDX^T$, where $D$ is a positive definite diagonal matrix with maximum diagonal entry as some $c$

One set of notes I saw just claimed that the maximum eigenvalue of $H$ is bounded above by $c||X||^2_2$. I believe that $||X||_2$ refers to the $l_2$ operator norm

I am struggling to prove this. Can anyone help me see how the claim is true? I know that the maximum eigenvalue is bounded by the operator norm. But I am unable to write $||H||$ in terms of $||X||$

Note: All Matrices here are Real.

asked Oct 10 '18 at 23:35

Vikash Balasubramanian

https://math.stackexchange.com/questions/252819/why-is-frobenius-norm-of-a-matrix-greater-than-or-equal-to-the-2-norm – Oct 10 '18 at 23:40
@RHowe, thanks. I totally missed the fact that H was normal – Vikash Balasubramanian Oct 10 '18 at 23:52

Bound on eigenvalues of matrices of the form $XDX^T$

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