Game Development Reference

In-Depth Information

31

0

=

.

(3.81)

V

2

31

0

These systems yield the solutions

1

1

1

3

V

=

a

1

V

=

b

,

(3.82)

2

−

where the scalars
a
and
b
are arbitrary nonzero constants.

In general, the eigenvalues of a matrix, given by the roots of its characteristic

polynomial, are complex numbers. This means that the corresponding eigen-

vectors can also have complex entries. A type of matrix that is guaranteed to

have real eigenvalues and therefore real eigenvectors, however, is the symmetric

matrix.

Definition 3.24.
An
nn

for all

i
and
j
. That is, a matrix whose entries are symmetric about the main diagonal

is called symmetric.

matrix
M
is
symmetric
if and only if

×

M

=

M

ij

ji

The eigenvalues and eigenvectors of symmetric matrices possess the proper-

ties given by the following two theorems.

Theorem 3.25.
The eigenvalues of a symmetric matrix M having real entries

are real numbers.

Proof.
Let
λ
be an eigenvalue of the matrix
M
, and let
V
be a corresponding ei-

genvector such th
at

MVV
. Multiplying both sides of this equation on the left

=

λ

V
gives us

by the row vector

T

T

T

V MV

=

V

λ λ

V

=

V V
,

(3.83)

where the overbar denotes complex conjugation, which for vectors and matrices

is performed componentwise. Since the product of a complex number
abi

+

and

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