Decomposition of a matrix

Question

Let $M$ be a real matrix of size $m$ x $n$. Show that it is possible to find a vector $a$ of size $n$ and a vector $b$ of size $m$ such that

$M a=\sigma b$,

$\|a\|_{2}=1$,

$\|b\|_{2}=1$ and

$\sigma=\|M\|_{2}$.

I have tried using the definition of the matrix norm induced by vectors in vain. Can someone help me ?

Answer 1

Hint

By definition (or by a trivial theorem from any equivalent definition), $$ \|M\|_2=\sup_{\|x\|=1}\|Mx\|_2\ , $$ the set $\ S^{n-1}=\big\{x\,\big|\,\|x\|_2=1\big\}\subset\mathbb{R}^n\ $ is compact, and the function $\ x\mapsto\|M\|_2\ $ is continuous, so it attains its supremum on $\ S^{n-1}\ $.