1. The document discusses different types of matrices that are important in power systems analysis, including square, overdetermined, underdetermined, triangular, diagonal, and identity matrices. 2. It also discusses special types of matrices such as null, symmetric, skew-symmetric, real, pure-imaginary, Hermitian, skew-Hermitian, orthogonal, and unitary matrices. 3. Examples of each matrix type are provided with representative element values.

1. 1
CHAPTER 1
MATRICES IN POWER SYSTEMS
1.1 General Matrices:
The use of matrix algebra for the formulation and solution of complicated
power system problems has become increasingly important since the
adventure of digital computers.
1.1.1 Types of Matrices
1) Square matrix: (no. of rows = no. of columns)
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎣
⎡
−
−
=
6220
4721
5102
4321
A
2) Overdetermine matrix: (no. of rows > no. of columns)
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
⎡
−
−
−
=
3193
6220
4721
5102
4321
1024
A
3) Underdetermine matrix: (no. of rows < no. of columns)
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎣
⎡
−
−
−
=
362201
147210
951022
343214
A

2. 2
4) Upper triangular matrix: (aij = 0 for all i > j)
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎣
⎡
=
6000
4700
5100
4321
A
5) Lower triangular matrix: (aij = 0 for all i < j)
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎣
⎡
−
=
7000
0721
0002
0001
A
6) Diagonal matrix: (aij = 0 for all i ≠ j)
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎣
⎡
−
=
7000
0700
0080
0001
A
7) Unit (U) or Identity (I) matrix: (aij = 0 for all i ≠ j & aij = 1 for all i = j)
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎣
⎡
=
1000
0100
0010
0001
A
1.1.2 Special Matrices
1) Null if A = A
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎣
⎡
=
0000
0000
0000
0000
A
2) Symmetric if A = At

3. 3
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎣
⎡
=
6454
4713
5102
4321
A
3) Skew-Symmetric if - A = At
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎣
⎡
−
−
−
−−−
=
0454
4013
5102
4320
A
4) Real if A = A*
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎣
⎡
++−−+
+++−−
++++
++++
=
06020200
04070201
05010002
04030201
jjjj
jjjj
jjjj
jjjj
A
5) Pure-imaginary if A = -A*
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎣
⎡
++−+
+++−
++++
++++
=
60202000
40702010
50100020
40302010
jjjj
jjjj
jjjj
jjjj
A
6) Hermition if A = (A*
)t
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎣
⎡
+−++
+++−
−−++
−+−+
=
06445544
44071133
55110022
44332201
jjjj
jjjj
jjjj
jjjj
A
7) Skew-Hermition if A = -(A*
)t
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎣
⎡
+−−−+
+−−−
−−+
+−−−+−
=
0445544
4401133
5511022
4433220
jjj
jjj
jjj
jjj
A

4. 4
8) Orthogonal if Qt
Q = I
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎣
⎡
−−
−−
−−
=
2835.06157.05519.04858.0
3040.02291.06650.06425.0
6122.04734.04966.03931.0
6726.05868.00812.04434.0
Q
(where Q is an orthonormal basis for the range of the square matrix A)
9) Unitary if (A*
)t
A = U
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
+−−−
−−+−=
35.05.035.05.01
35.05.035.05.01
111
3
1
jj
jjA
References:
[1] G.W. Stagg and A.H. El-Abiad, "Computer Methods in Power System
Analysis", McGraw-Hill, New York, 1968.