I have a matrix A, and two matrix norms: || . ||_alpha and || . ||_beta. I know that if I change A to A' then ||A||_alpha < ||A'||_alpha
My question is, in this case, is this true if I use ||.||_beta instead of ||.||_alpha ?
2026-03-16 12:08:31.1773662911
I have two matrix norms, the first grows, what happens with the other one?
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No it's not true.
At best, what you know is that both norms are equivalent, that is there exist constants $s>0$ and $t>0$ such that $$s\|.\|_\alpha \leq \|.\|_\beta\leq t\|.\|_\alpha$$ This is a consequence of those norms being over finite-dimensional vector spaces. But you can't expect any monotonicity.