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