# Matrix Spectral Radius and Induced Matrix Norms

Let $A$ be a matrix, and $\rho(A)$ be its spectral radius, $\|A\|_p$ be an norm induced from vector $p$-norm.

(1) When $\rho(A)=\|A\|_2$ or $\rho(A)=\|A\|_1$, does $\|A\|_1=\|A\|_2$?

(2) If the answer to question (1) is in the negative, does making $A$ normal or Hermitian imply $\|A\|_1=\|A\|_2$?

(3) What is an example of positive matrix $A$ where $\rho(A)<\|A\|_2$ or $\rho(A)<\|A\|_1$

• Could you define specifically the norms you are referring to? For some norms such an example does not exist, while it is obvious that there are norms with many examples. Commented Aug 2, 2014 at 21:19
• @AmitaiYuval: I have completely modified my question.
– Hans
Commented Aug 3, 2014 at 1:21
• It is worth noting that $A$ is normal $\implies \|A\|_2 = \rho(A)$. The converse does not generally hold. Commented Aug 3, 2014 at 3:40
• @Omnomnomnom: Exactly. That is why I word question (2) the way it is.
– Hans
Commented Aug 3, 2014 at 4:54
• @AmitaiYuval: Could you please state which norm gives $\rho(A)=\|A\|$ for all matrix?
– Hans
Commented Aug 3, 2014 at 4:57

(1): No. Take $$A = \pmatrix{-1&1\\1&1}$$ We find $\|A\|_2 = \rho(A) = \sqrt 2$, but $\|A\|_1 = 2$.
Conversely, take $$B = \pmatrix{2&1\\0&1}$$ We find $\|A\|_1 = \rho(A) = 2$, but $\|A\|_2 = \sqrt{3 + \sqrt5}$.
(3): In addition to the above example, take $$A = \pmatrix{1&1\\\epsilon&1}$$ For some small $\epsilon>0$. We have $\|A\|_2 \approx \sqrt{\frac{3 + \sqrt5}{2}}, \|A\|_1 = 2, \rho(A) \approx 1$.
Or, for a more particular example, take $$A = \pmatrix{2&2\\1&2}$$ We have $\rho(A) = 2 + \sqrt 2$, $\|A\|_1 = 4$, and $\|A\|_2 = \sqrt{\frac{13+3 \sqrt{17}}{2}}$