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Question

(a) If ||A|| is an operator norm, prove that

I=1,\| 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=maxx=1Ax\|A\|=\underset{\|x\|=1}{\text{max}}\|Ax\| and A=IA=I we obtain:

I=maxx=1IxI=maxx=1xI=maxx=11I=1\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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