My Math Forum Proof concerning matrix norm.

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April 6th, 2016, 09:24 AM   #1
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Proof concerning matrix norm.

Hi, I am struggling with following proof, could you give me some hint or some information which would help me in proving following inequality?
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 June 1st, 2016, 07:25 AM #2 Senior Member   Joined: Mar 2015 From: New Jersey Posts: 1,449 Thanks: 106 Definition* ||A||$\displaystyle _{\infty}$=max sum of absolute values of row elements. Theorem ||AB||$\displaystyle _{\infty} \leq$ ||A||$\displaystyle _{\infty}$||B||$\displaystyle _{\infty}$ Proof Consider A & B 2x2 and elements are the absolute values: Sum each row of AB: 1st row sum= a11(b11+b12) + a12(b21+b22) 2nd row sum= a21(b11+b12) + a22(b21+b22) Assume (b11+b12)$\displaystyle \leq$(b21+b22) 1st row sum$\displaystyle \leq$ (a11+a12)(b21+b22) 2nd row sum$\displaystyle \leq$ (a21+a22)(b21+b22) Assume (a11+a12)$\displaystyle \leq$(a21+a22) 1st row sum$\displaystyle \leq$ (a21+a22)(b21+b22) 2nd row sum$\displaystyle \leq$ (a21+a22)(b21+b22) In general, max of sum of absolute values of elements of rows of AB $\displaystyle \leq$ (max of sum of absolute values of elements of rows of A)( max of sum of absolute values of elements of rows of B). ||AB||$\displaystyle _{\infty} \leq$ ||A||$\displaystyle _{\infty}$||B||$\displaystyle _{\infty}$ * https://en.wikipedia.org/wiki/Matrix_norm Thanks from manus Last edited by zylo; June 1st, 2016 at 07:32 AM.

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