A question concerning matrix norms

[kenleung5e28]

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.

 

[morphism]

Yes. Just write down the definition of ||A||_a and show that it's equal to ||B||_b.

 

[kenleung5e28]

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 ||x||_a and ||x||_b may be of different values.

 

[radou]

Are you referring to operator norms here?