0 copyright c 2000 siam buy online

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: rgument shows that it’s also impossible to have p < q, so p = q. Thus it is proved that if A ∼ B, then A and B have the same inertia. Conversely, if A and B have in= ertia (p, j, s), then the argument that produced (7.6.15) yields A ∼ E ∼ B. == 77 The fact that inertia is invariant under congruence is also a corollary of a deeper theorem stating that the eigenvalues of A vary continuously with the entries. The argument is as follows. Assume A is nonsingular (otherwise consider A + I for small ), and set X(t) = tQ + (1 − t)QR for t ∈ [0, 1], where C = QR is the QR factorization. Both X(t) and Y (t) = XT (t)AX(t) are nonsingular on [0, 1], so continuity of eigenvalues insures that no eigenvalue Y (t) can cross the origin as t goes from 0 to 1. Hence Y (0) = CT AC has the same number of positive (and negative) eigenvalues as Y (1) = QT AQ, which is similar to A. Thus CT AC and A have the same inertia. Copyright c 2000 SIAM Buy online from SIAM http://www.ec-securehost.com/SIAM/ot71.html Buy from AMAZON.com 570 Chapter 7 Eigenvalues and Eigenvectors http://www.amazon.com/exec/obidos/ASIN/0898714540 Example 7.6.5 Taylor’s theorem in n says that if f is a smooth real-valued function defined on n , and if x0 ∈ n×1 , then the value of f at x ∈ n×1 is given by f (x) = f (x0 ) + (x − x0 )T g(x0 ) + (x − x0 )T H(x0 )(x − x0 ) 3 + O( x − x0 ), 2 where g(x0 ) = ∇f (x0 ) (the gradient of f evaluated at x0 ) has components gi = ∂f /∂xi , and where H(x0 ) is the Hessian matrix whose entries are It is illegal to print, duplicate, or distribute this material Please report violations to [email protected] x0 given by hij = ∂ 2 f /∂xi ∂xj x0 . Just as in the case of one variable, the vector D E x0 is called a critical point when g(x0 ) = 0. If x0 is a critical point, then Taylor’s theorem shows that (x − x0 )T H(x0 )(x − x0 ) governs the behavior of f at points x near to x0 . This observation yields the following conclusions regarding local maxima or minima....
View Full Document

{[ snackBarMessage ]}

Ask a homework question - tutors are online