When are the eigenvalues of a a and aa strictly

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Unformatted text preview: independent set? O C 7.1.2. Without doing an eigenvalue–eigenvector computation, determine which of the following are eigenvectors for ⎛ ⎞ −9 −6 −2 −4 ⎜ −8 −6 −3 −1 ⎟ A=⎝ ⎠, 20 15 8 5 32 21 7 12 and for those which are eigenvectors, identify the associated eigenvalue. ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ −1 1 −1 0 ⎜ 1⎟ ⎜ 0⎟ ⎜ 0⎟ ⎜ 1⎟ (a) ⎝ ⎠ . (b) ⎝ ⎠ . (c) ⎝ ⎠ . (d) ⎝ ⎠. 0 −1 2 −3 1 0 2 0 68 In fact, this result was the motivation behind the original development of Gerschgorin’s circles. Copyright c 2000 SIAM Buy online from SIAM http://www.ec-securehost.com/SIAM/ot71.html Buy from AMAZON.com 7.1 Elementary Properties of Eigensystems http://www.amazon.com/exec/obidos/ASIN/0898714540 501 7.1.3. Explain why the eigenvalues of triangular and diagonal matrices ⎛ ⎞ ⎛ t11 t12 · · · t1n λ1 0 · · · 0 ⎜ 0 t22 · · · t2n ⎟ ⎜ 0 λ2 · · · 0 T=⎜ . . .. . . . ⎟ and D = ⎜ . .. ⎝. ⎝. . . .⎠ .. . . . . . . . 0 · · · tnn 0 0 0 ⎞ ⎟ ⎟ ⎠ · · · λn It is illegal to print, duplicate, or distribute this material Please report violations to meyer@ncsu.edu are simply the diagonal entries—the tii ’s and λi ’s. 7.1.4. For T = A 0 B C , prove det (T − λI) = det (A − λI)det (C − λI) to B conclude that σ A C 0 D E = σ (A) ∪ σ (C) for square A and C. 7.1.5. Determine the eigenvectors of D = diag (λ1 , λ2 , . . . , λn ) . In particular, what is the eigenspace associated with λi ? T H 7.1.6. Prove that 0 ∈ σ (A) if and only if A is a singular matrix. ⎛n 1 ⎜1 n = ⎜1 1 ⎝. . .. IG 7.1.7. Explain why it’s apparent that An×n R Y . 1 . 1 1 1 n . . . 1 ··· ··· ··· .. . ··· ⎞ 1 1 ⎟ 1⎟ .⎠ . . doesn’t n have a zero eigenvalue, and hence why A is nonsingular. 7.1.8. Explain why the eigenvalues of A∗ A and AA∗ are real and nonneg2 2 ative for every A ∈ C m×n . Hint: Consider Ax 2 / x 2 . When are ∗ ∗ the eigenvalues of A A and AA strictly positive? P O C 7.1.9. (a) If...
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