Suppose $\displaystyle D\in\mathbb{C}^{nxn}$ is a diagonal and positif definite matrix.

Show that the induced matrix norm from the vector norm $\displaystyle \Vert x\Vert_{D}$ is $\displaystyle \Vert A\Vert_{D}=\max\sigma(D^{1/2}AD^{-1/2})$, that is the maximum singular value of $\displaystyle D^{1/2}AD^{-1/2}$ .

(Note : $\displaystyle D^{1/2}$ denote non negatif definite matrix $\displaystyle X$ satisfy $\displaystyle X^{2}=D$)

thx for the answer.