# Help verifying the norm of the resolvent of a matrix

I'm reading a document where it is said that if $$A=\begin{pmatrix}0 & 1\\0 & 0 \end{pmatrix}$$ then the norm of the resolvent for $$z \neq 0$$ is given by $$\|R(z,A)\|= \frac{\sqrt{2}}{\sqrt{1+2|z|^2-\sqrt{1+4|z|^2}}}.$$ I think that if $$z\neq 0$$ then $$R(z,A)=(A-zI)^{-1}=\begin{pmatrix}-1/z & -1/z^2\\0 & -1/z \end{pmatrix}$$ and because of that $$\|R(z,A)\|=\frac{\sqrt{2|z|^2+1}}{|z|^2}.$$

Am I wrong?.

• The spectral norm of $A$ coincides with the largest spectral value of $A$, i.e., the largest eigenvalue of $A^\top A$. – gerw Apr 8 at 19:50