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568
Chapter 7
Eigenvalues and Eigenvectors
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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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