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April 25th, 2015, 01:05 AM   #1
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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.
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April 25th, 2015, 03:46 AM   #2
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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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