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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
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7.2 Diagonalization by Similarity Transformations
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
13x2 + 10xy + 13y 2 = 72. It is illegal to print, duplicate, or distribute this material
Please report violations to email@example.com By rotating the xy -coordinate system counterclockwise through an angle of 45◦ D
E u v y T
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
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
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