1. ## Matrix and eigenvalues

A set of linear difference equation is given by Uj+1 = A Uj + bj, where the non singular NxN matrix A has N distinct nonzero eigenvalues lambda(i) and the corresponding eigenvector x(i). Show that ||A^j||<=M where M is a positive number independent of j, is equivalent to |lambda(i)|<=1 for all i=1,2...N

Anyone can help?

2. ## Re: Matrix and eigenvalues

There are several different matrix norms- which do you intend here? Or do you want to prove this is true for all matrix norms?

Since A has distinct eigenvalues, A= BDB^(-1) where D is a diagonal matrix having the eigenvalues of A on its Diagonal and B has the eigenvectors of A as its columns. Then A^2= (A)(A)= (BAB^(-1))(BAB^(-1))= BD^2B^(-1), A^3=A(A^2)= (BDB^(-1))(BD^2B^(-1)= BD^3B^(-1), etc. so that for any positive integer, n, A^n= BD^nB^(-1). For any norm, ||XY||<= ||X||||Y|| so ||A^n||<= ||B||||A||||B^(-1)||<= ||A||.

3. ## Re: Matrix and eigenvalues

I think it is any norm.
The ||B|| and ||B^-1|| in your solution, do they cancel out each other?