Is every matrix norm compatible with a vector norm?
in $\mathbb{C}^n$, I know every vector norm induces a matrix norm, and an induced matrix norm is compatible to its dedicated vector norm. So for every vector norm there exists a matrix norm, where the matrix norm is compatible with the vector norm. But is the other implication true too?
Is there for every matrix norm a vector norm it is compatible with?
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$\begingroup$Yes. This is the result of theorem 5.7.13 on p.373 of Horn and Johnson's Matrix Analysis (2/e).
A (submultiplicative) matrix norm $\|\cdot\|_M$ is said to be compatible with a vector norm $\|\cdot\|_v$ if $\|Ax\|_v\le\|A\|_M\|x\|_v$ for every matrix $A\in M_n(\mathbb C)$ and every vector $v\in\mathbb C^n$. Now, given any matrix norm $\|\cdot\|_M$, we pick a fixed but arbitrary nonzero vector $y$ and define a vector norm $\|\cdot\|_v$ by $\|x\|_v=\|xy^\ast\|_M$. Then $\|\cdot\|_M$ is compatible with $\|\cdot\|_v$ because$$ \|Ax\|_v=\|Axy^\ast\|_M\le\|A\|_M\|xy^\ast\|_M=\|A\|_M\|x\|_v. $$
$\endgroup$ 3 $\begingroup$The Frobenius norm $\|A \|_F = \bigl(\sum \limits_{i, j} A_{i,j}^2 \bigr)^{1/2}$ is not compatible with any vector norm. You can see this by looking at $\|I\|_F = \sqrt{n}$, which can not be compatible with a vector norm, as $\|x\| = \|I \cdot x\|$ holds for any vector norm.
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