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

by OC2716258

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**preview**shows page 1. to view the full**4 pages of the document.**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 k∈R. 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:R2→R.

(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

1+bx1x2+cx2

2

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.

1

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