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Question

# (a) If ||A|| is an operator norm, prove that$\| I \| = 1,$where I is an identity matrix. (b) Is there a vector norm that induces the Frobenius norm as an operator norm? Why or why not?

Solution

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#### (a)

From $\|A\|=\underset{\|x\|=1}{\text{max}}\|Ax\|$ and $A=I$ we obtain:

\begin{align*} \|I\|&=\underset{\|x\|=1}{\text{max}}\|Ix\|\\ \|I\|&=\underset{\|x\|=1}{\text{max}}\|x\|\\ \|I\|&=\underset{\|x\|=1}{\text{max}} 1\\ \|I\|&=1 \end{align*}

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