Databases Reference
In-Depth Information
whose
ij
th element is the cofactor
A
ji
. The inverse of a matrix
A
, denoted by
A
−
1
,isgiven
by
1
|
A
|
(
A
−
1
=
A
)
(B.20)
Notice that for the inverse to exist the determinant has to be nonzero. If the determinant for
a matrix is zero, the matrix is said to be singular. The method we have described here works
well with small matrices; however, it is highly inefficient if
N
becomes greater than 4. There
are a number of efficient methods for inverting matrices; see the topics in the Further Reading
section for details.
Corresponding to a squarematrix
A
of size
N
×
N
are
N
scalar values called the
eigenvalues
of
A
. The eigenvalues are the
N
solutions of the equation
|
λ
−
A
| =
I
0. This equation is called
the
characteristic equation
.
Example B.2.1:
Let us find the eigenvalues of the matrix
45
21
|
λ
I
−
A
| =
0
λ
45
21
=
0
−
0
0
λ
(λ
−
4
)(λ
−
1
)
−
10
=
0
λ
1
=−
1
λ
2
=
6
(B.21)
The eigenvectors
V
k
of an
N
×
N
matrix are the
N
vectors of dimension
N
that satisfy the
equation
A
V
k
=
λ
k
V
k
(B.22)
Further Reading
1.
The subject of matrices is covered at an introductory level in a number of textbooks. A
good one is
Advanced Engineering Mathematics
, by E. Kreyszig [
290
].
2.
Numerical methods for manipulating matrices (and a good deal more) are presented
in
Numerical Recipes in C
, by W.H. Press, S.A. Teukolsky, W.T. Vetterling, and B.P.
Flannery [
182
].
