- #1

Simfish

Gold Member

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## Homework Statement

Using the fact that [tex]||A||_2 = \sqrt { \rho ( A^* A )}[/tex], prove that

[tex]||A||_2 \leq \sqrt { ||A||_1 ||A||_\infty }[/tex]. This is an easy estimate to

find in practice for an upper bound on [tex]||A||_2[/tex].

## Homework Equations

## The Attempt at a Solution

Or, in other words, the magnitude of the largest eigenvalue of A*A <= the product of the maximum row sum and the maximum column sum.

So I've been trying to proceed and I'm not totally sure what to do. One thing I've figured: equality exists when the matrix is a diagonal matrix (so the entire matrix's values occur where they contribute to the eigenvalues).

There is an inequality [tex]||A||_2 < \sqrt{m} * ||A||_\infty[/tex] but I doubt this would help since the inequality would contain an extra factor of [tex]\sqrt{m}[/tex] (and an extra factor of sqrt(n) too, once you accounted for the 1-norm)