If I have a square matrix in $\mathbf{A} \in \mathbb{R}^{n \times n}$, I want to find another diagonal matrix $\mathbf{D} \in \mathbb{R}^{n \times n}$ that minimizes the residual $ \min_\mathbf{D}  \mathbf{AD} ^2 $, where the norm here is the induced norm $\max_{x\neq 0} \frac{\mathbf{Ax_2}}{\mathbf{x_2}}$. Is there a currentlyknown closed form for this optimization?
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The case of the $2$norm may well not have a closed form. However, in the Frobenius norm the problem has a trivial answer: $D_A = \mathrm{diag}(A)$. Since $\ A \_2 \leq \ A \_F \leq \sqrt{n} \ A \_2$, one then has that $$ \frac{1}{\sqrt{n}} \ A  D_A \ \leq \min_{D} \ A  D \ \leq \ A  D_A \, $$ which, at the least, gives a bound. 

