MAT444solution_7

# MAT444solution_7 - Solution 7 Sec 5.2 2 For the matrix A M...

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Solution 7 Sec. 5.2 2. For the matrix ) R ( M A nxn , test A for diagonal and ability and if A is diagonalizable, find an invertible matrix Q and a diagonal matrix D such that Q- 1 AQ=D (d) 3 6 6 0 5 8 0 4 7 Ans.: (d) 1 , 3 , 3 0 ] 32 ) 5 )[( 3 ( = = + λ ) 1 , 0 , 0 ( and ) 0 , 1 , 1 ( r eigenvecto 0 6 6 0 8 8 0 4 4 ) 3 ( = = I A ) 0 , 2 , 1 ( r eigenvecto 4 6 6 0 4 8 0 4 8 ) ( = = + I A = 0 1 0 2 0 1 1 0 1 Q 3. (c) For the linear operators T on a vector space V, test T for diagonalizability and if T is a diagonalizable, find a basis β for V such that [ ] T is a diagonal matrix. Ans.: (c) = 0 1 0 0 0 1 T = 0 0 1 0 1 0 T = 2 0 0 1 0 0 T [] = 2 0 0 0 0 1 0 1 0 T γ [] ⇒ = = = 0 ) 1 )( 2 ( 0 2 0 0 0 1 0 1 ) 1 T det( 2 P269 condition 1 fails to hold T is not a diagonalizable matrix.

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12. Let T be an invertible linear operator on a finite-dimensional vector space V. (a) Recall that for any eigenvalue λ of T, 1 is an eigenvalue of T -1 . Prove that the eigenvalue of T corresponding to is the same as the eigenspace T -1 of corresponding to 1 . (b) Prove that if T is diagonalizable, then T -1 is diagonalizable. Ans.: (a) For each eigenvector ν and correspond eigenvalue of linear operator T.
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MAT444solution_7 - Solution 7 Sec 5.2 2 For the matrix A M...

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