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Spectral norm

The spectral norm of a matrix $A^{m\times n}$ is equal to its largest singular value:

$\|A\|_2 = \max_{\|x\|_2=1} \|Ax\|_2 = \sigma_{\max}$

Alternate definition

We show that $\|A\|_2 = \max_{\|x\|=\|y\|=1} x^T A y$ , where $x$ and $y$ are vectors.

Proof:

Let $A=U^T\Sigma V$ be its SVD.

$\begin{eqnarray*} \|A\|_2 &=& \max_{\|x\|_2=\|y\|_2=1} (xU)^T \Sigma (Vy) \\ &=& \max_{\|x'\|_2=\|y'\|_2=1} x'^T \Sigma y' \\ &=& \max_{\|x'\|_2=\|y'\|_2=1} \sum_i \sigma_i x_i y_i \\ &\leq& \sigma_{\max} \sqrt{\|x\|_2^2 \|y\|_2^2} = \sigma_{\max} \end{eqnarray*}$

The equality holds when $x'=y'= [1,0,\cdots,0]^T$ (assuming $\sigma_1$ is the largest singular value).

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Spectral norm

Alternate definition

Proof:

Power method

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Table of Contents

Spectral norm

Alternate definition

Proof:

Power method

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