$||A||_2 = s u p_{x!=0} ||Ax||/||x||=s u p_{x!=0} sqrt((<Ax, Ax>)/(<x,x>))=s u p_{x!=0} sqrt((<A^T Ax,x>)/(<x,x>))=$ posto $y=U^T x$
$=s u p_{y!=0} sqrt((<A^T A Uy, Uy>)/(<Uy,Uy>))=$ U è ortogonale
$=s u p_{y!=0} sqrt((<A^T A Uy, Uy>)/(<y,y>))=s u p_{y!=0} sqrt((<U^T A^T A U y, y>)/(<y,y>))=s u p_{y!=0} sqrt((<diag (lambda_1,...,lambda_n) y, y>)/(<y,y>))=s u p_{y!=0} sqrt((sum_{i=1}^{n} lambda_i y_i ^2)/(sum_{i=1}^{n} y_i^2))=sqrt(max(lambda_1,...,lambda_n))$