Hint recall exercise 5914 d e 7225 the real schur form

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Unformatted text preview: f P−1 ⎜.⎟ . ∗ = ⎜ .∗ ⎟, where the yi ’s are rows ⎝y ⎠ t O C Y∗ ∗ ∗ ∗ and Y∗ is (n − t) × n, then {y1 , y2 , . . . , yt } is a set of linearly independent left-hand eigenvectors associated with {λ1 , λ2 , . . . , λt } , respec∗ ∗ tively (i.e., yi A = λi yi ). 7.2.8. Let A be a diagonalizable matrix, and let ρ( ) denote the spectral radius (recall Example 7.1.4 on p. 497). Prove that limk→∞ Ak = 0 if and only if ρ(A) < 1. Note: It is demonstrated on p. 617 that this result holds for nondiagonalizable matrices as well. 7.2.9. Apply the technique used to prove Schur’s triangularization theorem (p. 508) to construct an orthogonal matrix P such that PT AP is upper triangular for A = Copyright c 2000 SIAM 13 16 −9 −11 . Buy online from SIAM http://www.ec-securehost.com/SIAM/ot71.html Buy from AMAZON.com 522 Chapter 7 Eigenvalues and Eigenvectors http://www.amazon.com/exec/obidos/ASIN/0898714540 7.2.10. Verify the Cayley–Hamilton theorem for A = 1 8 −8 −4 −11 8 −4 −8 5 . It is illegal to print, duplicate, or distribute this material Please report violations to meyer@ncsu.edu Hint: This is the matrix from Example 7.2.1 on p. 507. 7.2.11. Since each row sum in the following symmetric matrix A is 4, it’s clear that x = (1, 1, 1, 1)T is both a right-hand and left-hand eigenvector associated with λ = 4 ∈ σ (A) . Use the deflation technique of Example 7.2.6 (p. 516) to determine the remaining eigenvalues of ⎛ ⎞ 1021 ⎜0 2 1 1⎟ A=⎝ ⎠. 2110 1102 D E 7.2.12. Explain why AGi = Gi A = λi Gi for the spectral projector Gi associated with the eigenvalue λi of a diagonalizable matrix A. T H 7.2.13. Prove that A = cn×1 dT×n is diagonalizable if and only if dT c = 0. 1 IG R 0 7.2.14. Prove that A = W Z is diagonalizable if and only if Ws×s and 0 Zt×t are each diagonalizable. 7.2.15. Prove that if AB = BA, then A and B can be simultaneously triangularized by a unitary similarity transformation—i.e., U∗...
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