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# 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 λ of L can be computed by finding the roots of the characteristic polynomial p A ( λ ) = det[ λI n- A ] . Example. Consider a diagonal matrix in the form D = λ 1 ··· λ 2 ··· . . . . . . . . . . . . ··· λ n = ⇒ λI n- D = λ- λ 1 ··· λ- λ 2 ··· . . . . . . . . . . . . ··· λ- λ n Since these are both diagonal matrices, the characteristic polynomial is simply the product of the diagonal elements: p D ( λ ) = det[ λI n- D ] = ( λ- λ 1 )( λ- λ 2 ) ··· ( λ- λ n ) . Hence the eigenvalues of a diagonal matrix D are the elements λ 1 , λ 2 ..., λ 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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lecture_37 - MA 265 LECTURE NOTES MONDAY APRIL 14 Similar...

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