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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
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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
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Last edited by zylo; June 1st, 2016 at 07:32 AM.
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