lecture_37 - MA 265 LECTURE NOTES: MONDAY, APRIL 14 Similar...

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Unformatted text preview: MA 265 LECTURE NOTES: MONDAY, APRIL 14 Similar Matrices Review. Let V = R n , and say that L : V V is a linear operator. We have seen that there is an n n matrix A such that L ( u ) = A u ; this is the standard matrix representing L . Similarly, we have seen that the eigenvalues of L can be computed by finding the roots of the characteristic polynomial p A ( ) = det[ I n- A ] . Example. Consider a diagonal matrix in the form D = 1 2 . . . . . . . . . . . . n = I n- D = - 1 - 2 . . . . . . . . . . . . - n Since these are both diagonal matrices, the characteristic polynomial is simply the product of the diagonal elements: p D ( ) = det[ I n- D ] = ( - 1 )( - 2 ) ( - n ) . Hence the eigenvalues of a diagonal matrix D are the elements 1 , 2 ..., n on the main diagonal. Diagonalization. Let A and B be n n matrices. We say that A and B are similar matrices if there exists a nonsingular n n matrix P such that B = P- 1 AP . Moreover, if B = D can be chosen to be a diagonal matrix, we say that A is diagonalizable ....
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This note was uploaded on 03/11/2010 for the course MA 261A 0026100 taught by Professor ... during the Spring '10 term at Purdue University Calumet.

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lecture_37 - MA 265 LECTURE NOTES: MONDAY, APRIL 14 Similar...

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