Consider any vector induced norm (need not be just p-norm). I need to prove or disprove that for any real symmetric matrix A and vector induced norm $||.||$,
$$ ||A|| = \rho (A) $$
where $\rho(A)$ is the spectral radius (largest eigen value in modulus).
Please provide any example if it can be disproved. Thank you.
I am disappointed with the comments. However I found an example to disprove.
$$ A=\begin{pmatrix} 2&-1\\ -1&3\\ \end{pmatrix} $$
It is easy verify that 1norm is 4 but 2norm is less than 4.