Page 311 - Advanced engineering mathematics

9.3 Some Special Types of Matrices 291

For n = 2, the quadratic form is
2 2

a jk z j z k = a 11 z 1 z 1 + a 12 z 1 z 2 + a 21 z 1 z 2 + a 22 z 2 z 2 .
j=1 k=1
The two middle terms are called mixed product terms, involving z j and z k with j = k.
If the quadratic form is real, then all of the numbers involved are real. In this case the
conjugates play no role and this quadratic form can be written
2 2

a jk x j x k = a 11 x 1 x 1 + a 12 x 1 x 2 + a 21 x 1 x 2 + a 22 x 2 x 2
j=1 k=1
2
2
= a 1 x + (a 12 + a 21 )x 1 x 2 + a 22 x .
1
2
As we have seen previously (in the discussion immediately preceding Lemma 9.1), we can let
t
A =[a jk ] and write the complex quadratic form as Z AZ, where
⎛ ⎞
z 1
z 2
⎜ ⎟
Z = ⎜ . ⎟.
⎜ ⎟
.
⎝ . ⎠
z n
t
If all the quantities are real, we usually write this as X AX. In fact, any real quadratic form can
be written in this way, with A a real symmetric matrix. We will illustrate this process.
EXAMPLE 9.16
Consider the real quadratic form

14
x 1 2 2

x 1 x 2 = x + 3x 1 x 2 + 4x 2 x 1 + 2x 2
1
32 x 2
2
2
= x + 7x 1 x 2 + 2x .
1
2
We can write the same quadratic form as

1 7/2

x 1 2 2
x 1 x 2 = x + 7x 1 x 2 + 2x 2
7/2 2 x 2
1
in which A is a symmetric matrix.
This is important in developing a standard change of variables that is used to simplify
quadratic forms by eliminating cross product terms.
THEOREM 9.13 Principal Axis Theorem
Let A be a real symmetric matrix with distinct eigenvalues λ 1 ,···λ n . Then there is an orthog-
onal matrix Q such that the change of variables X = QY transforms the quadratic form
n n a ij x i x j to
j=1 k=1
n
2
λ j y .
j
j=1

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October 14, 2010 14:49 THM/NEIL Page-291 27410_09_ch09_p267-294