The matrix norm induced by a vector norm ||.||_a is defined by
||A||_a = \sup_{x \neq 0} \frac{||Ax||_a}{||x||_a}. In showing the equality of matrix norms under the condition I've posted last time, I don't know how to deal with the denominator appeared in the denominator inside the supremum as...
I wonder given ||Ax||_{a} = ||Bx||_{b} for any x \in \mathbb{R}^n, is it true that ||A||_{a} = ||B||_{b}, where ||.||_{a}, ||.||_{b} are two vector norms and the matrix norms are induced by the corresponding vector norms?
Thanks in advance.
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