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?