Bounds on the eigenvalues of $\bf W D W^\top$

Question

Given a diagonal matrix $\bf D$ with diagonal entries $d_{ii} \in [0,1]$ and a matrix $\bf W$ with singular values $\sigma_i ({\bf W}) \in [0,1]$, can it be proven that the eigenvalues of $\bf W D W^\top$ lie in the interval $[0,1]$?

It's worth noting that $\bf W D W^\top$ is positive semidefinite, ensuring real non-negative eigenvalues.

Any insights or proofs would be greatly appreciated.

Answer 1

The answer is YES. To prove it, you can prove that $$\mathbf I - \mathbf W \mathbf D\mathbf W^\intercal \succeq 0$$

Indeed,

• $\mathbf W\mathbf W^{\intercal} - \mathbf W \mathbf D\mathbf W^\intercal = \mathbf W \underbrace{\left(\mathbf I - \mathbf D\right)}_{\succeq 0}\mathbf W^\intercal \succeq 0$

• The eigenvalues of $\mathbf W\mathbf W^\intercal$ are the singular values of $\mathbf W$ squared. Since these values lie in $[0, 1]$ then the eigenvalues of $\mathbf W\mathbf W^\intercal$ will also lie in $[0, 1]$ and that proves that, $$\mathbf I - \mathbf W\mathbf W^\intercal \succeq 0.$$