Lecture 34 - Mar 30

Lecture 34 - Mar 30 - Monday March 30 Lecture 34:...

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Monday March 30 Lecture 34: Diagonalizable matrices II . (Refers to section 6.1, 6.2, 6.3) Expectations: 1. Diagonalize diagonalizable matrices. 34.1 Example Show that the following matrix is diagonalizable. Then diagonalize it. Produce the required diagonalizing matrix P and the diagonal matrix. 4 2 3 3 We easily find two eigenvalues λ 1 = 1 and λ 2 = 6 and corresponding eigenvectors x 1 = (2, 3) and x 2 = (1, 1). Setting P = 2 1 3 1 We get P -1 = 1/5 1/5 3/3 2/5 Then verify that P -1 A P = 1 0 0 6 34.2 Example Suppose A is a diagonalizable matrix. Then we know there exist an invertible matrix P and a diagonal matrix D such that P -1 A P = D . Show that for any positive integer n, A n = P D n P -1 .
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Solution: Simply note that A n = ( P D P -1 )( P D P -1 )( P D P -1 )...( P D P -1 ) = P D n P -1 . 34.2.1 Remark It is easily verified that if D is a diagonal matrix then D n is a diagonal matrix whose entries on the diagonal are those in D to the power of n . (Since if D = [ a ij ] is diagonal, D 2 = [< r i , c j >]. Then if i j , < ri , cj > = 0, and if i = j then < r i , c j > = < r i , c i > = < r i , r i > = a ii 2 . ) 34.3 Examples 1) Show that the given matrix A is diagonalizable. Then find a matrix P such that P -1 A P = D . Verify that the non-zero entries of
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This note was uploaded on 06/10/2010 for the course MATH 136 taught by Professor All during the Winter '08 term at Waterloo.

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Lecture 34 - Mar 30 - Monday March 30 Lecture 34:...

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