My Math Forum Eigenvalue Problem

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 September 16th, 2013, 08:29 AM #1 Newbie   Joined: Sep 2013 Posts: 2 Thanks: 0 Eigenvalue Problem If A is the 4 by 4 matrix of ones, what are the eigenvalues and the determinant of A - I? How to solve this without expanding?
 September 16th, 2013, 09:15 AM #2 Senior Member   Joined: Jun 2013 From: London, England Posts: 1,316 Thanks: 116 Re: Eigenvalue Problem The determinant will be 0, as you will have 1 -1 + 1 - 1 times the determinant of the same 3x3 matrix with all 1's. Any vector (a, b, c, d) will become (a+b+c+d, a+b+c+d, a+b+c+d, a+b+c+d) under the action of this matrix. If this is an eigenvector with eigenvalue k, then a+b+c+d = ka = kb =kc = kd. Hence a = b = c = d and k = 4.
 September 16th, 2013, 11:16 AM #3 Newbie   Joined: Sep 2013 Posts: 2 Thanks: 0 Re: Eigenvalue Problem I think you wrote about the eigenvalues and determinant of the matrix A. But I need those of the matrix A-I. I understand A has eigenvalues 4,0,0,0 and Determinant is 0. But how do I get the results for A-I from this?
 September 16th, 2013, 12:00 PM #4 Senior Member   Joined: Jun 2013 From: London, England Posts: 1,316 Thanks: 116 Re: Eigenvalue Problem The matrix A - I acting on the vector (a, b, c, d) will produce the vector (b+c+d, a+c+d, a+b+d, a+b+c). If this is an eigenvector then either: a = b = c =d and the eigenvalue k = 3 or k = -1 and a = -(b+c+d) This can be seen by solving the system of simultaneous equations. The determinant of A is -9.

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