lecture_37 (dragged) 2

lecture_37 (dragged) 2 - MA 36600 LECTURE NOTES FRIDAY...

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Unformatted text preview: MA 36600 LECTURE NOTES: FRIDAY, APRIL 23 3 It is unclear whether we can find a matrix T such that T−1 A T is a diagonal matrix because we only have one eigenvector. Consider instead the matrix ￿ ￿ ￿ ￿￿ ￿ 1 −1 0 1 0 1 0 −1 T= =⇒ T= = . −1 −1 11 −1 −1 −1 Consider the product ￿ ￿￿ ￿￿ ￿￿ ￿￿ ￿￿ ￿ 1 0 1 −1 1 0 1 0 2 1 21 −1 J = T AT = = = . −1 −1 1 3 −1 −1 −1 −1 −2 −3 02 We claim that this is the best we can do i.e., there does not exist a matrix T such that T−1 A T = D is a diagonal matrix. Assume to the contrary, that such a matrix does exist. Then D = 2 I would be a diagonal matrix with 2’s along the diagonal. Then we have ￿ ￿ T−1 A T = D =⇒ A = T D T−1 = T (2 I) T−1 = 2 T I T−1 = 2 I = D. This implies that A = D is a diagonal matrix, which is clearly a contradiction. Hence A is not diagonalizable. ...
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This note was uploaded on 11/30/2011 for the course MATH 366 taught by Professor Edraygoins during the Spring '09 term at Purdue.

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