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Anonymous Tue 15 Jun 20:48:01 2021 No.13281774 View Reply Original Report

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I have a question about spectral norms of matrices. Specifically why does:
$\|A\|_2^2=\max\{|\lambda|:\lambda\in\sigma(A^\intercalA)\}=\max\{|\lambda|:\lambda\in\sigma(AA^\intercal)\}=\|A^\intercal\|_2^2.$

Anonymous

Anonymous Tue 15 Jun 2021 20:50:40 No.13281776 Report

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$\|A\|_2^2=\max\{|\lambda|:\lambda\in\sigma(A^\intercal A)\}=\max\{|\lambda|:\lambda\in\sigma(AA^\intercal)\}=\|A^\intercal\|_2^2$

Anonymous

Anonymous Tue 15 Jun 2021 21:30:19 No.13281861 Report

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the spectral norm is trying to characterize the maximum that a matrix can scale a vector, right?
$\max_{\mathbf{x}} \frac{\|\mathbf{A} \mathbf{x}\|^2}{\|\mathbf{x}\|^2} = \max_{\mathbf{x}}\frac{\langle \mathbf{A} \mathbf{x}, \mathbf{A} \mathbf{x}\rangle}{\langle \mathbf{x}, \mathbf{x}\rangle} = \max_{\mathbf{x}} \frac{\mathbf{x}^H \mathbf{A}^H \mathbf{A} \mathbf{x}}{ \mathbf{x}^H \mathbf{x}}$
$\mathbf{A}^H \mathbf{A}$ is conjugate-symmetric and positive-semidefinite, so its spectrum is composed of real non-negative eigenvalues
the largest ratio between the L2 norm of the transformed vector $\mathbf{A} \mathbf{x}$ and the input vector $\mathbf{x}$ is achieved when the input vector is an eigenvector of the largest eigenvalue $\mathbf{A} \mathbf{x}$ .

i have to think some more about the second half of the equation...

Anonymous

Anonymous Tue 15 Jun 2021 21:34:02 No.13281870 Report

Quoted By: >>13281922

the spectral norm is trying to characterize the maximum that a matrix can scale a vector, right?
$\max_{\mathbf{x}} \frac{\|\mathbf{A} \mathbf{x}\|^2}{\|\mathbf{x}\|^2} = \max_{\mathbf{x}}\frac{\langle \mathbf{A} \mathbf{x}, \mathbf{A} \mathbf{x}\rangle}{\langle \mathbf{x}, \mathbf{x}\rangle} = \max_{\mathbf{x}} \frac{\mathbf{x}^H \mathbf{A}^H \mathbf{A} \mathbf{x}}{ \mathbf{x}^H \mathbf{x}}$
$\mathbf{A}^H \mathbf{A}$ is conjugate-symmetric and positive-semidefinite, so its spectrum is composed of real non-negative eigenvalues
the largest ratio between the L2 norm of the transformed vector $\mathbf{A} \mathbf{x}$ and the input vector $\mathbf{x}$ is achieved when the input vector is an eigenvector of the largest eigenvalue $\mathbf{A}^H \mathbf{A}$ .

i have to think some more about the second half of the equation...

Anonymous

Anonymous Tue 15 Jun 2021 21:50:55 No.13281922 Report

Quoted By:

>>13281870
Exactly that. Thank you for looking into this. Much appreciated.

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