hw_sol(5_2) - A 2 ? What is the determinant of ( A-1 ) T ?...

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Section 5.2 8 Suppose A = uv T is a column times a row (a rank-1 matrix). (a) By multiplying A times u , show that u is an eigenvector. What is λ ? Solution. Au = ( uv T ) u = u ( v T u ) = ( v T u ) · u , so by definition, u is an eigenvector with the associated eigenvalue v T u . (b) What are the other eigenvalues of A (and why)? Solution. The other eigenvalues are zero. This is because, if x is an eigenvector, then λx = Ax = ( uv T ) x = ( v T x ) · u, which means x is a multiple of u . But we know that the eigenvectors associated to different eigenvectors are linearly indep, so v T x and λ should be zero. (c) Compute trace ( A ) from the sum on the diagonal and the sum of λ ’s. Solution. The trace of A is v T u , which is also the sum of the eigenvalues. 10 Suppose A has eigenvalues 1, 2, 4. What is the trace of
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Unformatted text preview: A 2 ? What is the determinant of ( A-1 ) T ? Proof. (Assume the eigenvalues have multiplicity 1.) The trace of A 2 is 1 2 + 2 2 + 3 2 = 14. This is because, suppose A is diagonalizable, then S-1 AS = diag(1 , 2 , 3) and S-1 A 2 S = diag(1 1 , 2 2 , 3 2 ). Similarly det( A-1 ) T = det( A-1 ) = 1 1 + 1 2 + 1 3 = 11 6 . 32 Diagonalize A and compute S k S-1 to prove this formula for A k : A = " 2 1 1 2 # has A k = 1 2 " 3 k + 1 3 k-1 3 k-1 3 k + 1 # . 36 If A = S S-1 , diagonalize the block matrix B = " A 0 2A # . Find its eigenvalue and eigenvector matrices. Proof. S = " S S # . 1...
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