Norm of diagonal and orthogonal matrix

Question

I need your expertise in evaluating the following problem:

Let $D \in \mathbb{R}^{d \times d}$ be a diagonal matrix and let $V \in \mathbb{R}^{d \times d}$ be a orthogonal matrix, then it is know that:

• $\left| \left| D \right| \right| _2 = \max_{i \in [d]}\left\lbrace \lambda_i\right\rbrace$, where $\lambda_i$ is the $i^{th}$ eigenvalue of $D$ for every $i \in [n]$.
• $\left| \left| V \right| \right|_2 = 1 $

So what I want to know is the $l_2$ norm of $DV^{\top}$ and the $l_2$ norm of $V^{\top}D^{-1}$?

Answer 1

The norm is the same as the one of D because the map associated with V (or its inverse) is an isometry.