
Abstract Algebra Abstract Algebra Math Forum 
 LinkBack  Thread Tools  Display Modes 
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: 831 Thanks: 67 
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. 

Tags 
matrix, norm, proof 
Thread Tools  
Display Modes  

Similar Threads  
Thread  Thread Starter  Forum  Replies  Last Post 
Proof concerning eigenvalue and matrix norm.  hegendroffer  Abstract Algebra  2  June 2nd, 2016 04:30 AM 
Derivative of norm of a Gram matrix's diagonal  mnnejati  Calculus  0  April 6th, 2015 11:04 AM 
Matrix REF Proof  Student444  Linear Algebra  4  February 5th, 2012 11:30 PM 
Matrix norm optimization problem  onako  Linear Algebra  0  February 1st, 2012 01:59 AM 
Minimize the infinity norm of a matrix  ricconna  Linear Algebra  0  August 23rd, 2011 08:55 PM 