Page 336 - A Course in Linear Algebra with Applications

320 Chapter Nine: Advanced Topics

Therefore A is diagonalized by the orthogonal matrix

/ W2 W6 W3\
S = -1/^2 1A/6 1/^3 .
V 0 -2/V6 1/V3/

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The matrix 5 represents a rotation of axes. P u t X = 5 X ;
then X = SX' and

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r
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X A X = X') (5 A5)X = (X'fDX,
(
where D is the diagonal matrix with diagonal entries 0, 0, 3.
The equation of the quadric becomes

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X' DX' + (-l 2 -1)SX' = 0
or

V2 V®
This is a parabolic cylinder whose axis is the line with equa-
tions y' = y/Zx', z' = 0.

Definite quadratic forms
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Consider once again a quadratic form q = X AX in real
variables x±,..., x n, where A is a real symmetric matrix. In
some applications it is the sign of q that is significant.
The quadratic form q is said to be positive definite if q > 0
whenever 1 ^ 0 . Similarly, q is called negative definite if q < 0
whenever X ^ 0. If, however, q can take both positive and
negative values, then q is said to be indefinite. The terms
positive definite, negative definite and indefinite can also be
applied to a real symmetric matrix A, according to the behav-
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ior of the corresponding quadratic form q = X AX.
For example, the expression 2x 2 + 3y 2 is positive unless
x = 0 = y, so this is a positive definite quadratic form, while