# Questions tagged [matrix-norms]

This tag is for questions regarding the Matrix Norm, a vector norm in a vector space whose elements (vectors) are matrices (of given dimensions).

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### The norm $\|S-Q\|_F$ where $Q$ is orthogonal is minimised by $Q=I$

Problem: Suppose that $S$ is symmetric and semi-positive-definite. Let $\|\cdot \|_F$ be the Frobenius norm. Show that $$\|S-I \|_F \leq \|S-Q\|_F$$ for all orthogonal matrices $Q$, where $I$ is ...
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### Equality for matrix norms [duplicate]

Let $A \in \mathbb{C}^{n,n}$ invertible and we use a norm $\|\cdot\|$. Then it holds that $$\|A^{-1}\|= \max_{||x||=1} \|A^{-1}x\|=\max_{\|x\|=1}\frac{1}{\|Ax\|}.$$ I can see, why that holds for ...
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### Matrix norm exercises

Let $f,g:\mathbb{R}^{n\times n}\to \mathbb{R}$ be given by \begin{aligned} f(A) &= \max_{i=1,...,n}i\cdot \sum_{j=1}^{n}|a_{ij}|,\\ g(A) &= n\cdot \max_{i,j=1,...,n}|a_{ij}| \end{aligned}...