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Unformatted text preview: ⎟ ⎜ ⎟ . ⎟⎜ ⎟ ⎜ ⎟ . . . ⎟⎜ . ⎟ ⎜ ⎟ . . ⎟⎜ . ⎟ ⎜ ⎟ (k−3) ⎟ ⎜ ⎜ f (λi ) ⎟ .⎟ (k − 1)(k − 2)λi ⎟⎜ ⎟ ⎜ ⎟ ⎟⎜ ⎟ ⎜ ⎟ . ⎟⎜ ⎟ ⎜ ⎟ . . ⎟⎜ ⎟ ⎜ ⎟ . . ⎟⎜ . ⎟ ⎜ ⎟ . ⎟⎜ . ⎟ ⎜ ⎟ .⎟ ⎟⎜ ⎜ ⎟ ⎠⎝ ⎠ ⎝ ⎠ . . . . . αk−1 . λk−1 1 . . . k −1 λs The coeﬃcient matrix H can be proven to be nonsingular because the rows in each segment of H are linearly independent. The rows in the top segment of H are a subset of rows from a Vandermonde matrix (p. 185), while the nonzero portion of each succeeding segment has the form VD, where the rows of V are a subset of rows from a Vandermonde matrix and D is a nonsingular diagonal matrix. Consequently, Hx = f has a unique solution, and thus there is a unique polynomial p(z ) = α0 + α1 z + α2 z 2 + · · · + αk−1 z k−1 that satisﬁes the conditions in (7.9.16). This polynomial p(z ) is called the Hermite interpolation polynomial, and it has the property that f (A) = p(A). Copyright c 2000 SIAM Buy online from SIAM http://www.ec-securehost.com/SIAM/ot71.html Buy from AMAZON.com 608 Chapter 7 Eigenvalues and Eigenvectors http://www.amazon.com/exec/obidos/ASIN/0898714540 Example 7.9.5 It is illegal to print, duplicate, or distribute this material Please report violations to [email protected] Functional Identities. Scalar functional identities generally extend to the matrix case. For example, the scalar identity sin2 z + cos2 z = 1 extends to matrices as sin2 Z + cos2 Z = I, and this is valid for all Z ∈ C n×n . While it’s possible to prove such identities on a case-by-case basis by using (7.9.3) or (7.9.9), there is a more robust approach that is described below. Example 7.9.6 For two functions f1 and f2 from C into C and for a polynomial p(x, y ) in two variables, let h be the composition deﬁned by h(z ) = p f1 (z ), f2 (z ) . If An×n has eigenvalues σ (A) = {λ1 , λ2 , . . . , λs } with index (λi ) = ki , and if h is deﬁned at A,...
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