Paul horst usa 1935 along with faddeev and sominskii

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Unformatted text preview: λn ) be a diagonal real matrix such that λ1 < λ2 < · · · < λn , and let vn×1 be a column of real nonzero numbers. (a) Prove that if α is real and nonzero, then λi is not an eigenvalue for D + αvvT . Show that the eigenvalues of D + αvvT are in fact given by the solutions of the secular equation f (ξ ) = 0 defined by n 2 vi f (ξ ) = 1 + α . λ −ξ i=1 i For n = 4 and α > 0, verify that the graph of f (ξ ) is as depicted in Figure 7.1.4, and thereby conclude that the eigenvalues of D + αvvT interlace with those of D. Copyright c 2000 SIAM Buy online from SIAM http://www.ec-securehost.com/SIAM/ot71.html Buy from AMAZON.com 504 Chapter 7 Eigenvalues and Eigenvectors http://www.amazon.com/exec/obidos/ASIN/0898714540 1 ξ1 ξ2 It is illegal to print, duplicate, or distribute this material Please report violations to meyer@ncsu.edu λ1 λ2 1 ξ3 ξ4 λ3 λ4 λ4 + α D E Figure 7.1.4 −1 (b) Verify that (D − ξi I) v is an eigenvector for D + αvvT that is associated with the eigenvalue ξi . T H 7.1.23. Newton’s Identities. Let λ1 , . . . , λn be the roots of the polynomial p(λ) = λn + c1 λn−1 + c2 λn−2 + · · · + cn , and let τk = λk + λk + · · · + λk . n 1 2 Newton’s identities say ck = −(τ1 ck−1 + τ2 ck−2 + · · · + τk−1 c1 + τk )/k. Derive these identities by executing the following steps: n (a) Show p (λ) = p(λ) i=1 (λ − λi )−1 (logarithmic differentiation). −1 (b) Use the geometric series expansion for (λ − λi ) to show that for |λ| > maxi |λi |, n IG R 1 τ2 n τ1 = + 2 + 3 + ···. (λ − λi ) λλ λ Y P i=1 (c) Combine these two results, and equate like powers of λ. 69 7.1.24. Leverrier–Souriau–Frame Algorithm. Let the characteristic equation for A be given by λn + c1 λn−1 + c2 λn−2 + · · · + cn = 0, and define a sequence by taking B0 = I and trace (ABk−1 ) Bk = − I + ABk−1 for k = 1, 2, . . . , n. k Prove that for each k, trace (ABk−1 ) ck = − . k Hint: Use...
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