Suppose that $\Bbb R^m$ and $\Bbb R^n$ are equipped with norms $\|\cdot\|_b$ and $\|\cdot\|_a$ respectively. Show that the induced matrix norm $\|\cdot\|_{a,b}$ can be computed by the formula
$$\|A\|_{a,b} = \max\limits_{x\neq 0}\dfrac{\|Ax\|_b}{\|x\|_a}$$
and i use this definition for induced matrix
can anyone answer this question?
Hint: Note that $$ \frac{\|Ax\|_b }{\|x\|_a} = \left\| A\left(\frac{x}{\|x\|_a}\right)\right\|_b. $$ Now, show that $$ \begin{align*} \max\left\{\frac{\|Ax\|_b}{\|x\|_a} : x \neq 0\right\}&= \max\{\|Ax\|_b : \|x\|_a = 1\} \\ & = \max \{\|Ax\|_b: \|x\|_a \leq 1\} = \|A\|_{a,b} \end{align*} $$