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June 8th, 2012, 11:07 AM  #1 
Member Joined: Jan 2012 Posts: 51 Thanks: 1  Eigenvalues and Eigenvectors
Let A be an nxn invertible matrix and let ? be an eigenvalue of A. Show that 1/? is an eigenvalue of A^1. Can someone help me with this proof ? Sorry about my english, I'm brazilian. 
June 8th, 2012, 11:38 AM  #2 
Member Joined: Jan 2012 Posts: 51 Thanks: 1  Re: Eigenvalues and Eigenvectors
Well, i think i got the idead of the exercise above, but I'll let it here to anyone who wants to solve. Now I'm having trouble with this one: Let A be an nxn matriz with the propertie of A=A². Show that if ? is an eigenvalue, then it is 0 or 1.

June 8th, 2012, 03:01 PM  #3  
Global Moderator Joined: May 2007 Posts: 6,704 Thanks: 669  Re: Eigenvalues and Eigenvectors Quote:
? is an eigenvalue means A  ?I = 0. Let B = A^(1). I  ?B = BA  ?B = B(A  ?I) = BA  ?I = 0. Therefore 1/? is an eigenvalue for B.  
June 8th, 2012, 04:41 PM  #4  
Member Joined: Jan 2012 Posts: 51 Thanks: 1  Re: Eigenvalues and Eigenvectors Quote:
The proof shows 1/? is an eigenvalue of B.  
June 8th, 2012, 07:02 PM  #5 
Math Team Joined: Sep 2007 Posts: 2,409 Thanks: 6  Re: Eigenvalues and Eigenvectors
I wouldn't bother with the determinant. Simply note that if v is an eigenvector of A with eigenvalue , then . Now multiply both sides by .


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