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MATH 33AH Lecture Notes - Lecture 23: A Priori And A Posteriori, Diagonalizable Matrix, Orthogonal MatrixExam

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Math 33A Week 10
November 29 and December 1, 2016
Definiteness of Quadratic Forms
Given a quadratic form q(~x), we often care about the range of values the form might take.
A priori, we know a few things about the values of q. We always have q(~
0) = 0, and the
range of qis unbounded, since
q(k~x) = k2~x
for any scalar kR. Questions whose answer we don’t know, however, include whether or
not qtakes on any negative values, or whether qhas a nontrivial kernel. Figure 1 shows
plots of various forms q:R2R.
(a) A positive-definite form. (b) A negative-definite form. (c) An indefinite form.
(d) A positive semi-definite form. (e) A negative semi-definite form.
Figure 1: Plots of quadratic forms.
In these plots we see five distinct behaviors. The form in Figure 1a never takes on a
negative value, and the only vector in its kernel is the zero vector. That is, q(~x)>0 for
all nonzero vectors ~x. On the other hand, Figure 1b has the property that q(~x)<0 for all
nonzero vectors ~x. We say that these forms are positive-definite and negative-definite,
respectively. Figure 1c shows a form that takes on both positive and negative values, and
we say that such a form is indefinite. The forms in Figures 1d and 1e are similar to those
in Figures 1a and 1b, respectively, in that their range is of the form [0,) or (−∞,0]. The
difference, though, is that the forms in Figures 1d and 1e both have nontrivial kernels. We
say that the form in Figure 1d is positive semi-definite, meaning that q(~x)0 for all ~x,
but that there is some nonzero vector ~x so that q(~x) = 0. Similarly, the form in Figure 1e
is called negative semi-definite. Our goal now is to classify quadratic forms according to
these five categories. In particular, we want to take a quadratic form q(~x) = ax2
and determine which of these five terms is in effect1.
1Of course, there’s one form which doesn’t fall into any of these five categories: the form q(~x) = 0.
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