Unformatted text preview: A 2 ? What is the determinant of ( A1 ) 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 S1 AS = diag(1 , 2 , 3) and S1 A 2 S = diag(1 1 , 2 2 , 3 2 ). Similarly det( A1 ) T = det( A1 ) = 1 1 + 1 2 + 1 3 = 11 6 . 32 Diagonalize A and compute S Λ k S1 to prove this formula for A k : A = " 2 1 1 2 # has A k = 1 2 " 3 k + 1 3 k1 3 k1 3 k + 1 # . 36 If A = S Λ S1 , diagonalize the block matrix B = " A 0 2A # . Find its eigenvalue and eigenvector matrices. Proof. S = " S S # . 1...
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 Spring '08
 Wakefield
 Linear Algebra, Matrices, Signal Processing, Eigenvalues, Eigenvalue, eigenvector and eigenspace, Orthogonal matrix

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