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April 6th, 2016, 09:24 AM  #1 
Newbie Joined: Jun 2015 From: Warsaw Posts: 3 Thanks: 1  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?

June 1st, 2016, 07:25 AM  #2 
Senior Member Joined: Mar 2015 From: New Jersey Posts: 915 Thanks: 72 
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 Last edited by zylo; June 1st, 2016 at 07:32 AM. 

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