Biomedical Engineering Reference
In-Depth Information
The inverse of a square matrix
A
, denoted,
A
1
, is defined by
A
1
A
AA
1
Þ
The use of the matrix inverse is important in solving a set of simultaneous equations that
describe a system.
The transpose of matrix
A
is denoted as either
A
0
or
A
T
, where we interchange rows and
columns of
A
to form
A
T
as follows from Equation (A.1).
¼
¼
I
ð
A
:
6
a
11
a
21
. . .
a
n
1
a
12
a
22
. . .
a
n
2
A
T
=
a
1
m
a
2
m
. . .
a
nm
SIMULTANEOUS EQUATIONS AND MATRICES
A set of equations with unknown variables often results in the process of analyzing a
system, after applying interconnection laws or other techniques. At other times, a set of
equations with unknown parameters results in solving the coefficients of a differential
equation with initial conditions.
Suppose the following set of equations describes the currents in an electric circuit,
i
i
,
i
2
,
and
i
3
, after applying Kirchhoff's voltage law.
2
i
i
þ
0
i
¼
6
1
2
3
2
i
þ
3
i
þ
i
¼
0
ð
A
:
7
Þ
1
2
3
i
þ
5
i
4
i
¼
0
1
2
3
The constants multiplying the currents involve the resistors in the circuit, with the value of
6 due to the input to the circuit. In writing each equation in (A.7) we have included all vari-
ables, written in order, even if a variable is multiplied by 0.
Equation (A.7) is written in matrix form as
2
2
3
2
3
2
3
10
i
1
i
2
i
3
6
0
0
4
5
4
5
¼
4
5
231
ð
A
:
8
Þ
15
4
or
Ai
¼
F
ð
A
:
9
Þ
with appropriately defined matrices
2
3
2
3
2
3
2
10
i
1
i
2
i
3
6
0
0
4
5
,
i
4
5
, and
F
4
5
A
¼
231
¼
¼
15
4
To solve Equation (A.8) for the current vector
i
, we premultiply both sides of the equation by
the inverse
A
1
, that is
A
1
Ai
A
1
F
¼
ð
A
:
10
Þ
or
A
1
F
i
¼
ð
A
:
11
Þ
