0
$\begingroup$

It is true that $\rho(A)\le\|A\|_p$ but is there an example where the inequality is strict? ($\rho$ is the spectral radius i.e. $\max|$eigenvalues of $A|$)

Though I understand the definition of a norm of a matrix I don't know how to calculate it.

$\endgroup$
1
$\begingroup$

Take\begin{array}{rccc}A\colon&\mathbb{R}^2&\longrightarrow&\mathbb{R}^2\\&(x,y)&\mapsto&(x+y,y).\end{array}Its only eigenvalue is $1$ and therefore its spectral radius is $1$. But $A(0,1)=(1,1)$ and therefore $\|A\|\geqslant\sqrt2$ (actually, it is an equality).

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.