I'm doing a problem from Pugh's Real Mathematical Analysis, the Linear Algebra review chapter, and am given the matrix $S = \left(\begin{array}{cc} 1 & s \\ 0 & 1 \end{array} \right)$.
I have the calculate the norm and conorm of $S$ and have a hint to use polar form and $SS^T$.
I know that $SS^T=\left(\begin{array}{cc} 1+s^2 & s \\ s & 1 \end{array} \right)=OP(OP)^T=OP^2O^{-1}$, as $O$ is an orthogonal matrix and $P$ is a positive diagonal matrix. From this I can conclude that $SS^T$ is diagonizable and has the same eigenvalues as $P^2$.
Now, I'm thinking of using the $\infty$ metric, and calculating something of this sort: $||SS^T||=||P^2||=\sup\{|(x+s^2x+sy, sx+y)|:|(x,y)|=1\}=\sup\{|1+s^2|+|s|,|s|+1\}=\max\{|1+s^2|+|s|,|s|+1\}=1+s^2+|s|$ and so $||P||=||S||=\sqrt{1+s^2+|s|}$.
And similarly for the conorm.
Does this make sense? Am I approaching this correctly and if not how should I think about it?
Thank you for any help.