# Amazoncomexecobidosasin0898714540 573 it is illegal to

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Unformatted text preview: ng a change of variables (coordinates) Copyright c 2000 SIAM Buy online from SIAM http://www.ec-securehost.com/SIAM/ot71.html Buy from AMAZON.com 568 Chapter 7 Eigenvalues and Eigenvectors http://www.amazon.com/exec/obidos/ASIN/0898714540 y = QT x. This follows because A is symmetric, so there is an orthogonal matrix Q such that QT AQ = D = diag (λ1 , λ2 , . . . , λn ) , where λi ∈ σ (A) , and setting y = QT x (or, equivalently, x = Qy ) gives n T T T 2 λi yi . T x Ax = y Q AQy = y Dy = (7.6.14) It is illegal to print, duplicate, or distribute this material Please report violations to [email protected] i=1 This shows that the nature of the quadratic form is determined by the eigenvalues of A (which are necessarily real). The eﬀect of diagonalizing a quadratic form in this way is to rotate the standard coordinate system so that in the new coordinate system the graph of xT Ax = α is in “standard form.” If A is positive deﬁnite, then all of its eigenvalues are positive (p. 559), so (7.6.14) makes it clear that the graph of xT Ax = α for a constant α > 0 is an ellipsoid centered at the origin. Go back and look at Figure 7.2.1 (p. 505), and see Exercise 7.6.4 (p. 571). D E T H Example 7.6.4 Congruence. It’s not necessary to solve an eigenvalue problem to diagonalize a quadratic form because a congruence transformation CT AC in which C is nonsingular (but not necessarily orthogonal) can be found that will do the job. A particularly convenient congruence transformation is produced by the LDU factorization for A, which is A = LDLT because A is symmetric—see Exercise 3.10.9 (p. 157). This factorization is relatively cheap, and the diagonal entries in D = diag (p1 , p2 , . . . , pn ) are the pivots that emerge during Gaussian elimination (p. 154). Setting y = LT x (or, equivalently, x = (LT )−1 y ) yields IG R Y P n 2 pi y i . xT Ax = yT Dy = O C i=1 The inertia of a real-symmetric matrix A is deﬁned to be the triple (ρ, ν, ζ ) in which ρ, ν, and ζ are the respective number of positive, negative, and zero eigenvalues, cou...
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