Show that for any real matrix $A$, $||A||_2=\sup_{x\ne0, y\ne 0}\frac{|y^TAx|}{||y||_2||x||_2}$

Question

Show that for any real matrix $A$, $$||A||_2=\sup_{x\ne0, y\ne 0}\frac{|y^TAx|}{||y||_2||x||_2}$$

Here i only know that $||A\|_2=\max_{x\neq 0}\frac{\|Ax\|_2}{\|x\|_2}$

can some one suggest me book of where i get metrics on matrices and norm on the matrices

thank you so much....

Answer 1

RHS $\leq $ is easy. [Use: $|u^{T}v| \leq \|u\|\|v\|$]. For the other way take $y=Ax$. [Use: $|(Ax)^{T} Ax|=\|Ax\|^{2}$].