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Consider the definitions of matrix norm and subordinate matrix norm from

Matrix Norm set #2

and let $A$ a real matrix and $\|\cdot\|$ a vector norm over $\mathbb{C}^n$. Define \begin{eqnarray*} \|A\| & = & \sup_{v\in\mathbb{C}^n-\{0\}}\frac{\|Av\|}{\|v\|},\\ |A| & = & \sup_{v\in\mathbb{R}^n-\{0\}}\frac{\|Av\|}{\|v\|}. \end{eqnarray*}

  1. Show that $\|A\| = |A|$ if the vector norm is any of the norms $\|\cdot\|_p$, $p=1,2,\infty$. Is the result also true for the norms $\|\cdot\|_p$, for the remaining values of $p \geq 1$?
  2. Give an example of a matrix $A$ and a vector norm for which $|A| \leq \|A\|$.
  3. Prove that $\rho(A) < |A|$.

Please, somebody can help me with this question? Thanks in advance.

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    You can find an answer to 2. & 3. (which I presume should read $\rho(A) \le |A|$) in http://www.cs.berkeley.edu/~wkahan/MathH110/NormReal.pdf. – copper.hat Apr 22 '13 at 16:40
  • Also, I assume 2. should read $|A| < |A|$, otherwise the $p=1,2,\infty$ norms from 1. will suffice? – copper.hat Apr 22 '13 at 16:41

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