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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. Tags matrix, questions Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post silviatodorof Linear Algebra 2 March 22nd, 2015 05:28 AM omer1994 Linear Algebra 1 August 10th, 2014 02:37 PM imirish85 Linear Algebra 0 March 4th, 2011 10:22 AM excellents Linear Algebra 0 October 17th, 2009 08:12 AM

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