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
f (ξ ) = 1 + α
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
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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 [email protected] 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 .
Newton’s identities say ck = −(τ1 ck−1 + τ2 ck−2 + · · · + τk−1 c1 + τk )/k.
Derive these identities by executing the following steps:
(a) Show p (λ) = p(λ) i=1 (λ − λi )−1 (logarithmic diﬀerentiation).
(b) Use the geometric series expansion for (λ − λi )
to show that
for |λ| > maxi |λi |,
= + 2 + 3 + ···.
(λ − λi )
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.
Prove that for each k,
trace (ABk−1 )
ck = −
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