749 much of the subsequent development is accessible

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Unformatted text preview: Newton’s identities, and recall Exercise 7.1.10(a). O C 69 This algorithm has been rediscovered and modiﬁed several times. In 1840, the Frenchman U. J. J. Leverrier provided the basic connection with Newton’s identities. J. M. Souriau, also from France, and J. S. Frame, from Michigan State University, independently modiﬁed the algorithm to its present form—Souriau’s formulation was published in France in 1948, and Frame’s method appeared in the United States in 1949. Paul Horst (USA, 1935) along with Faddeev and Sominskii (USSR, 1949) are also credited with rediscovering the technique. Although the algorithm is intriguingly beautiful, it is not practical for ﬂoating-point computations. Copyright c 2000 SIAM Buy online from SIAM http://www.ec-securehost.com/SIAM/ot71.html Buy from AMAZON.com 7.2 Diagonalization by Similarity Transformations 505 http://www.amazon.com/exec/obidos/ASIN/0898714540 7.2 DIAGONALIZATION BY SIMILARITY TRANSFORMATIONS The correct choice of a coordinate system (or basis) often can simplify the form of an equation or the analysis of a particular problem. For example, consider the obliquely oriented ellipse in Figure 7.2.1 whose equation in the xy -coordinate system is 13x2 + 10xy + 13y 2 = 72. It is illegal to print, duplicate, or distribute this material Please report violations to meyer@ncsu.edu By rotating the xy -coordinate system counterclockwise through an angle of 45◦ D E u v y T H x IG R Figure 7.2.1 Y P into a uv -coordinate system by means of (5.6.13) on p. 326, the cross-product term is eliminated, and the equation of the ellipse simpliﬁes to become O C u2 v2 + = 1. 9 4 It’s shown in Example 7.6.3 on p. 567 that we can do a similar thing for quadratic equations in n . Choosing or changing to the most appropriate coordinate system (or basis) is always desirable, but in linear algebra it is fundamental. For a linear operator L on a ﬁnite-dimensional space V , the goal is to ﬁnd a basis B for V such that the matrix representation of L with respect to B is as simple as possible. Since diﬀerent matrix representations A and B of L are related by a similarity 70 transf...
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This document was uploaded on 03/06/2014 for the course MA 5623 at City University of Hong Kong.

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