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April 25th, 2015, 01:05 AM  #1 
Newbie Joined: Apr 2015 From: HK Posts: 2 Thanks: 0  3 questions about matrix
I have already try my best to do it, but still can't solve it Thank you for everyone who try to answer my question! 1. 2. 3. 
April 25th, 2015, 03:46 AM  #2 
Math Team Joined: Jan 2015 From: Alabama Posts: 3,264 Thanks: 902 
These all pretty straightforward problems. If you really cannot do them, you must be missing something basic. For the first, to say that v is an eigenvector of A with eigenvalue 7 means that Av= 7v. So A^2v= A(Av)= 7Av= 7(7v)= (7)^2v= 49v. Then A^3v= A(A^2v)= 49Av= 49(7v)= (7)^3v, etc Since Av= 7v, A^1(Av)= v= 7A^1v so that A^1v= (1/7)v. (A 5I)v= Av 5Iv= 7v 5v= 12v For the second, if a matrix, A, is "diagonalizable", then there exist matrix P such that A= PDP^1 where D is a diagonal matrix having the eigenvalues of A on its diagonal and P has the corresponding eigenvectors of A as its columns. The simplest way to find a high power of a (diagonalizable) matrix is to first diagonalize it. If A= PDP^1, then A^n= PD^nP^1. 

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