appendixa.matrixmethods

7/29/2019 AppendixA.MatrixMethods

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Appendix AMatrix Methods

A.1 REVIEW SUMMARY

A.1.1 Sets

A set of points is denoted by

S x1; x2; x3 A:1

This shows a set of three points, x1, x2, and x3. Some properties may be assigned to

the set, i.e.,

S fx1; x2; x3jx3 0g A:2

Equation (A.2) indicates that the last component of the set x3 = 0. Members of a set

are called elements of the set. If a point x, usually denoted by "xx, is a member of the

set, it is written as

"xx 2 S A:3

If we write:

"xx =2 S A:4

then point x is not an element of set S. If all the elements of a set S are also the

elements of another set T, then S is said to be a subset of T, or S is contained in T:

S & T A:5

Alternatively, this is written as

T ' S A:6The intersection of two sets S1 and S2 is the set of all points "xx such that "xx is an

element of both S1 and S2. If the intersection is denoted by T, we write:

T S1 \ S2 A:7

arcel Dekker, Inc. All Rights Reserved.


2/24

The intersection of n sets is

T S1 \ S2 \ . . . \ Sn \ni1Si A:8

The union of two sets S1 and S2 is the set of all points "xx such that "xx is an element of

either S1 or S2. If the union is denoted by P, we write:

P S1 [ S2 A:9

The union of n sets is written as:

P S1 [ S2 [ . . . [ Sn Uni1Si A:10

A.1.2 Vectors

A vector is an ordered set of numbers, real or complex. A matrix containing only one

row or column may be called a vector:

"xx

x1x2

xn

A:11

where x1, x2, . . ., xn are called the constituents of the vector. The transposed form is

"xx 0 jx1; x2; . . . ; xnj A:12

Sometimes the transpose is indicated by a superscript letter t. A null vector"

00 has allits components equal to zero and a sum vector "11 has all its components equal to 1.

The following properties are applicable to vectors

"xx "yy "yy "xx

"xx "yy "zz "xx "yy "zz

12 "xx 12 "xx

1 2 "xx 1 "xx 2 "xx

"00 "xx "00

A:13

Multiplication of two vectors of the same dimensions results in an inner or scalar

product:

"xx 0 "yy Xni1

xiyi "yy0

"xx

"xx 0 "xx j "xxj2

cos "xx 0 "yy

jxjjyj

A:14

where is the angle between vectors and |x| and |y| are the geometric lengths. Two

vectors "xx1 and "xx2 are orthogonal if:

"xx1 "xx02 0 A:15

Matrix Methods 713

02 by Marcel Dekker, Inc. All Rights Reserved.


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A.1.3 Matrices

1. A matrix is a rectangular array of numbers subject to certain rules of

operation, and usually denoted by a capital letter within brackets [A], a capital letter

in bold, or a capital letter with an overbar. The last convention is followed in this

book. The dimensions of a matrix indicate the total number of rows and columns.

An element aij lies at the intersection of row i and column j.2. A matrix containing only one row or column is called a vector.

3. A matrix in which the number of rows is equal to the number of columns is

a square matrix.

4. A square matrix is a diagonal matrix if all off-diagonal elements are zero.

5. A unit or identity matrix "II is a square matrix with all diagonal elements

=1 and off-diagonal elements = 0.

6. A matrix is symmetric if, for all values of i and j, aij = aji.7. A square matrix is a skew symmetric matrix ifaij = aji for all values of i

and j.

8. A square matrix whose elements below the leading diagonal are zero is

called an upper triangular matrix. A square matrix whose elements above the leading

diagonal are zero is called a lower triangular matrix.

9. If in a given matrix rows and columns are interchanged, the new matrix

obtained is the transpose of the original matrix, denoted by "AA 0.

10. A square matrix "AA is an orthogonal matrix if its product with its trans-

pose is an identity matrix:

"AA "AA 0 "II A:16

11. The conjugate of a matrix is obtained by changing all its complex ele-

ments to their conjugates, i.e., if

"AA 1 i 3 4i 5

7 2i i 4 3i

A:17

then its conjugate is

"AA 1 i 3 4i 5

7 2i i 4 3i A:18A square matrix is a unit matrix if the product of the transpose of the conjugate

matrix and the original matrix is an identity matrix:

"AA 0 "AA "II A:19

12. A square matrix is called a Hermitian matrix if every ij element is equal

to the conjugate complex ji element, i.e.,

"AA "AA 0 A:20

13. A matrix, such that:

"AA2 "AA A:21

is called an idempotent matrix.

714 Appendix A



4/24

14. A matrix is periodic if

"AAk1 "AA A:22

15. A matrix is called nilpotent if

"AAk 0 A:23

where k is a positive integer. Ifk is the least positive integer, then k is called the index

of nilpotent matrix.

16. Addition of matrices follows a commutative law:

"AA "BB "BB "AA A:24

17. A scalar multiple is obtained by multiplying each element of the matrix

with a scalar. The product of two matrices "AA and "BB is only possible if the number of

columns in "AA equals the number of rows in "BB.

If "AA is an m n matrix and "BB is n p matrix, the product "AA "BB is an m p

matrix wherecij ai1b1j ai2b2j ainbnj A:25

Multiplication is not commutative:

"AA "BB 6 "BB "AA A:26

Multiplication is associative if confirmability is assured:

"AA "BB "CC "AA "BB "CC A:27

It is distributive with respect to addition:

"AA "BB "CC "AA "BB "AA "CC A:28

The multiplicative inverse exists if jAj 6 0. Also,

"AA "BB 0 "BB 0 "AA 0 A:29

18. The transpose of the matrix of cofactors of a matrix is called an adjoint

matrix. The product of a matrix "AA and its adjoint is equal to the unit matrix multi-

plied by the determinant of A.

"AA "AAadj "IIjAj A:30

This property can be used to find the inverse of a matrix (see Example A.4).

19. By performing elementary transformations any nonzero matrix can be

reduced to one of the following forms called the normal forms:

Ir Ir 0Ir0

Ir 00 0

A:31

The number r is called the rank of matrix "AA. The form:

Ir 0

0 0 A:32is called the first canonical form of "AA. Both row and column transformations can be

used here. The rank of a matrix is said to be r if (1) it has at least one nonzero minor

of order r, and (2) every minor of "AA of order higher than r = 0. Rank is a nonzero

row (the row that does not have all the elements =0) in the upper triangular matrix.

Matrix Methods 715



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Example A.1

Find the rank of the matrix:

"AA

1 4 5

2 6 8

3 7 22

This matrix can be reduced to an upper triangular matrix by elementary row opera-

tions (see below):

"AA

1 4 5

0 1 1

0 0 12

The rank of the matrix is 3.

A.2 CHARACTERISTICS ROOTS, EIGENVALUES, AND

EIGENVECTORS

For a square matrix "AA, the "AA "II matrix is called the characteristic matrix; is

a scalar and "II is a unit matrix. The determinant jA Ij when expanded gives

a polynomial, which is called the characteristic polynomial of "AA and the equation

jA Ij 0 is called the characteristic equation of matrix "AA. The roots of the

characteristic equation are called the characteristic roots or eigenvalues.Some properties of eigenvalues are:

. Any square matrix "AA and its transpose "AA 0 have the same eigenvalues.

. The sum of the eigenvalues of a matrix is equal to the trace of the matrix

(the sum of the elements on the principal diagonal is called the trace of the

matrix).

. The product of the eigenvalues of the matrix is equal to the determinant of

the matrix. If

1; 2;. . .

; n

are the eigenvalues of "AA, then the eigenvalues of

k "AA are k1; k2; . . . ; kn

"AAm are m1 ; m2 ; . . . ;

mn

"AA1 are 1=1; 1=2; . . . ; 1=n

A:33

. Zero is a characteristic root of a matrix, only if the matrix is singular.

.

The characteristic roots of a triangular matrix are diagonal elements of thematrix.

. The characteristics roots of a Hermitian matrix are all real.

. The characteristic roots of a real symmetric matrix are all real, as the real

symmetric matrix will be Hermitian.

716 Appendix A



6/24

A.2.1 CayleyHamilton Theorem

Every square matrix satisfies its own characteristic equation:

If j "AA "IIj 1nn a1n1 a2

n2 an A:34

is the characteristic polynomial of an n n matrix, then the matrix equation:

"XXn a1 "XXn1 a2 "XXn2 an "II 0

is satisfied by "XX "AA

"AAn a1 "AAn1 a2 "AA

n2 an "II 0

A:35

This property can be used to find the inverse of a matrix.

Example A.2

Find the characteristic equation of the matrix:

"AA 1 4 23 2 2

1 1 2

and then the inverse of the matrix.

The characteristic equation is given by

1 4 2

3 2 2

1 1 2

0

Expanding, the characteristic equation is

3 52 8 40 0

then, by the CayleyHamilton theorem:

"AA2 5 "AA 8 "II 40 "AA1 0

40 "AA1 "AA2 5 "AA 8 "II

We can write:

40A

1

1 4 2

3 2 21 1 2

2

5

1 4 2

3 2 21 1 2 8

1 0 0

0 1 00 0 1 0

The inverse is

A1

0:05 0:25 0:3

0:2 0 0:2

0:125 0:125 0:25

This is not an effective method of finding the inverse for matrices of large dimen-

sions.

A.2.2 Characteristic Vectors

Each characteristic root has a corresponding nonzero vector "xx which satisfies the

equation j "AA "IIj "xx 0. The nonzero vector "xx is called the characteristic vector or

eigenvector. The eigenvector is, therefore, not unique.

Matrix Methods 717



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A.3 DIAGONALIZATION OF A MATRIX

If a square matrix "AA ofn n has n linearly independent eigenvectors, then a matrix "PP

can be found so that

"PP1 "AA "PP A:36

is a diagonal matrix.The matrix "PP is found by grouping the eigenvectors of "AA into a square matrix,

i.e., "PP has eigenvalues of "AA as its diagonal elements.

A.3.1 Similarity Transformation

The transformation of matrix "AA into P1 "AA "PP is called a similarity transformation.

Diagonalization is a special case of similarity transformation.

Example A.3

Let "AA

2 2 3

2 1 6

1 2 0

Its characteristics equation is

3 2 21 45 0

5 3 3 0

The eigenvector is found by substituting the eigenvalues:

7 2 3

2 4 6

1 2 3

x

y

z

0

0

0

As eigenvectors are not unique, by assuming that z 1, and solving, one eigenvector

is

1; 2; 1t

Similarly, other eigenvectors can be found. A matrix formed of these vectors is

"PP

1 2 3

2 1 0

1 0 1

and the diagonalization is obtained:

"PP1 "AA "PP 5 0 00 3 0

0 0 3

This contains the eigenvalues as the diagonal elements.

718 Appendix A



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A.4 LINEAR INDEPENDENCE OR DEPENDENCE OF VECTORS

Vectors "xx1; "xx2; . . . ; "xxn are dependent if all vectors (row or column matrices) are of the

same order, and n scalars 1; 2; . . . ; n (not all zeros) exist such that:

1 "xx1 2 "xx2 3 "xx3 n "xxn 0 A:37

Otherwise they are linearly independent. In other words, if vector "xxK 1 can bewritten as a linear combination of vectors x1; "xx2; . . . ; "xxn, then it is linearly depen-

dent, otherwise it is linearly independent. Consider the vectors:

"xx3

4

2

5

"xx1

1

0:5

0

"xx2

0

0

1

then

"xx3 4 "xx1 5 "xx2

Therefore, "xx3 is linearily dependent on "xx1 and "xx2.

A.4.1 Vector Spaces

If "xx is any vector from all possible collections of vectors of dimension n, then for any

scalar , the vector "xx is also of dimension n. For any other n-vector "yy, the vector

"xx "yy is also of dimension n. The set of all n-dimensional vectors are said to form a

linear vector space En. Transformation of a vector by a matrix is a linear transfor-

mation:

"AA "xx "yy "AA "xx "AA "yy A:38

One property of interest is

"AA "xx 0 A:39

i.e., whether any nonzero vector "xx exists which is transformed by matrix "AA into a

zero vector. Equation (A.39) can only be satisfied if the columns of "AA are linearly

dependent. A square matrix whose columns are linearly dependent is called a sin-

gular matrix and a square matrix whose columns are linearly independent is called a

nonsingular matrix. In Eq. (A.39) if "xx "00, then columns of "AA must be linearly

independent. The determinant of a singular matrix is zero and its inverse does not

exist.

A.5 QUADRATIC FORM EXPRESSED AS A PRODUCT OF

MATRICES

The quadratic form can be expressed as a product of matrices:

Quadratic form "xx 0A "xx A:40

where

"xx

x1x2x3

"AA

a11 a12 a13a21 a22 a23a31 a32 a33

A:41

Matrix Methods 719



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Therefore,

"xx 0A "xx x1 x2 x3 a11 a12 a13a21 a22 a23

a31 a32 a33

x1

x2

x3

a11x

2

1 a22x

2

2 a33x

2

3 2a12x; x2 2a23x2x3 2a13x1x3

A:42

A.6 DERIVATIVES OF SCALAR AND VECTOR FUNCTIONS

A scalar function is defined as

y ffi fx1; x2; . . . ; xn A:43

where x1; x2; . . . ; xn are n variables. It can be written as a scalar function of an n-

dimensional vector, i.e., y f "xx, where "xx is an n-dimensional vector:

"xx

x1x2

xn

A:44In general, a scalar function could be a function of several vector variables, i.e.,

y f "xx; "uu; "pp, where "xx; "uu, and "pp are vectors of various dimensions. A vector function

is a function of several vector variables, i.e., "yy f "xx; "uu; "pp.

A derivative of a scalar function with respect to a vector variable is defined as

@f

@x

@f=@x1@f=@x2

@f=@xn

A:45The derivative of a scalar function with respect to a vector ofn dimensions is a vector

of the same dimension. The derivative of a vector function with respect to a vector

variable x is defined as

@f=@x @f1=@x1 @f1=@x2 @f1=@xn@f2=@x2 @f2=@x2 @f2=@xn

@fm=@x1 @fm=@x2 @fm=@xn

@f1=@x1

T

@f2=@x2T

@fm=@xnT

A:46

If a scalar function is defined as

s Tf "xx "uu "pp

1 f1 "xx; "uu; "pp 2 f2 "xx; "uu; "pp m fm "xx; "uu; "ppA:47

then @s=@ is

@s

@

f1 "xx; "uu; "pp

f2 "xx; "uu; "pp

::

fm "xx; "uu; "pp

f"xx; "uu; "pp A:48

720 Appendix A



10/24

and @s=@x is

@s@x

1@f1

@x1 2

@f2

@x1 . . . m

@fm

@x1

1@f1

@x2 2

@f2

@x2 . . . m

@fm

@x2

. . . . . .

1@f1@xn

2@f2@xn

. . . m@fm@xn

@f1

@x1

@f2

@x1 . . .

@fm

@x1@f1

@x2

@f2

@x2 . . .

@fm

@x2

. . . . . .

@f1@xn

@f2@xn

. . . @fm@xn

1

2::

m

A:49

Therefore,

@s

@x

@f

@x

T

A:50

A.7 INVERSE OF A MATRIX

The inverse of a matrix is often required in the power system calculations, though it

is rarely calculated directly. The inverse of a square matrix "AA is defined so that

"AA1 "AA "AA "AA1 "II A:51

The inverse can be evaluated in many ways.

A.7.1 By Calculating the Adjoint and Determinant of the Matrix

"AA1 "AAadj

jAjA:52

Example A.4

Consider the matrix:

"AA

1 2 3

4 5 6

3 1 2

Its adjoint is

"AAadj 4 1 3

10 7 6

11 5 3

and the determinant of "AA is equal to 9.

Matrix Methods 721



11/24

Thus, the inverse of "AA is

"AA1

4

9

1

9

1

3

10

9

7

9

2

311

9

5

9

1

3

A.7.2 By Elementary Row Operations

The inverse can also be calculated by elementary row operations. This operation is as

follows:

1. A unit matrix of n n is first attached to the right side of matrix n n

whose inverse is required to be found.

2. Elementary row operations are used to force the augmented matrix so

that the matrix whose inverse is required becomes a unit matrix.

Example A.5

Consider a matrix:

"AA 2 6

3 4

It is required to find its inverse.

Attach a unit matrix of 2 2 and perform the operations as shown:

2 6

3 4

1 00 1

R12 1 33 4

12 00 1

R2 3R1 1 30 5

1

20

3

21

R1

5

3R2

1 0

0 5

2

5

3

53

21

R2 1

5

1 0

0 1

2

5

3

53

10

1

5

Thus, the inverse is

"AA1

2

5

3

53

10

1

5

Some useful properties of inverse matrices are:

The inverse of a matrix product is the product of the matrix inverses taken in

reverse order, i.e.,

"AA "BB "CC1 "CC1 "BB1 "AA1 A:53

The inverse of a diagonal matrix is a diagonal matrix whose elements are the respec-

tive inverses of the elements of the original matrix:

722 Appendix A



12/24

A11B22

C33

1

1

A111

B221

C33

A:54

A square matrix composed of diagonal blocks can be inverted by taking the inverse

of the respective submatrices of the diagonal block:

block A

block B

block C

1

block A1

block B1

block C1

A:55

A.7.3 Inverse by PartitioningMatrices can be partitioned horizontally and vertically, and the resulting submatrices

may contain only one element. Thus, a matrix "AA can be partitioned as shown:

"AA

a11 a12 a13 a14a21 a22 a23 a24a31 a32 a33 a34

a41 a42 a43 a44

"AA1 "AA2"AA3 "AA4

A:56

where

"AA

a11 a12 a13a21 a22 a23a31 a32 a33

A:57

"AA2

a14a24a34

"AA3 a41 a42 a43 "AA4 a44 A:58

Partitioned matrices follow the rules of matrix addition and subtraction. Partitioned

matrices "AA and "BB can be multiplied if these are confirmable and columns of "AA androws of "BB are partitioned exactly in the same manner:

"AA1122"AA1221

"AA2112"AA2211

"BB1123"BB1221

"BB2113"BB2211

"AA11 "BB11 "AA12 "BB21 "AA11 "BB12 "AA12 "BB22"AA21 "BB11 "AA22 "BB21 "AA21 "BB12 "AA22 "BB22

A:59

Example A.6

Find the product of two matrices A and B by partitioning:

"AA

1 2 3

2 0 1

1 3 6

"BB

1 2 1 0

2 3 5 1

4 6 1 2

Matrix Methods 723



13/24


14/24

"AA 003 1

7

!1 2 1 3

1 2

17 17

"AA 004 1 2 1 3

1 2 0

3 4 !1

1

7

"AA1

2

7

12

7

9

71

7

8

7

6

7

1

7

1

7

1

7

A.8 SOLUTION OF LARGE SIMULTANEOUS EQUATIONSThe application of matrices to the solution of large simultaneous equations consti-

tutes one important application in the power systems. Mostly, these are sparse

equations with many coefficients equal to zero. A large power system may have

more than 3000 simultaneous equations to be solved.

A.8.1 Consistent Equations

A system of equations is consistent if they have one or more solutions.

A.8.2 Inconsistent Equations

A system of equations that has no solution is called inconsistent, i.e., the following

two equations are inconsistent:

x 2y 4

3x 6y 5

A.8.3 Test for Consistency and Inconsistency of Equations

Consider a system of n linear equations:

a11x1 a12x2 a1nx1 b1

a21x1 a22x2 A2nx2 b2

an1x1 an2x2 amnxn bn

A:64

Form an augmented matrix "CC:

"CC "AA; "BB

a11 a12 a1n b1a21 a22 a2n b2

an1 an2 ann bn

A:65

Matrix Methods 725



15/24

The following holds for the test of consistency and inconsistency:

. A unique solution of the equations exists if: rank of "AA = rank of "CC n,

where n is the number of unknowns.

. There are infinite solutions to the set of equations if: rank of "AA = rank of"CC r, r < n.

. The equations are inconsistent if rank of"

AA is not equal to rank of"

CC.

Example A.8

Show that the equations:

2x 6y 11

6x 20y 6z 3

6y 18z 1

are inconsistent.The augmented matrix is

"CC "AA "BB

2 6 0 11

6 20 6 3

0 6 18 1

It can be reduced by elementary row operations to the following matrix:

2 6 0 11

0 2 6 300 0 0 91

The rank of A is 2 and that of C is 3. The equations are not consistent.

The equations (A.64) can be written as

"AA "xx "bb A:66

where "AA is a square coefficient matrix, "bb is a vector of constants, and "xx is a vector of

unknown terms. If "AA is nonsingular, the unknown vector "xx can be found by

"xx "AA1 "bb A:67

This requires calculation of the inverse of matrix "AA. Large system equations are not

solved by direct inversion, but by a sparse matrix techniques.

Example A.9

This example illustrates the solution by transforming the coefficient matrix to an

upper triangular form (backward substitution). The equations:

1 4 6

2 6 3

5 3 1

x1x2x3

2

1

5

726 Appendix A



16/24

can be solved by row manipulations on the augmented matrix, as follows:

1 4 6

2 6 3

5 3 1

2

1

5

R2 2R1

1 4 6

0 2 9

5 3 1

2

3

5

R3 5R1

1 4 60 2 9

0 17 29

23

5

R3

17

2R2

1 4 60 2 9

0 0 47:5

23

20:5

Thus,

47:5x3 20:5

2x2 9x3 3

x1 4x2 6x3 2

which gives

"xx

1:179

0:442

0:432

A set of simultaneous equations can also be solved by partitioning:

a11; ; a1k a1m; ; a1n:: ::

ak1; ; akk akm; ; akn

am1; ; amk amm; ; amn:: ::

an1; ; ank anm; ; ann

x1

xk

xm

xn

b1

bk

bm

bn

A:68

Equation (A.68) is horizontally partitioned and rewritten as

"AA1 "AA2"AA3 "AA4

"XX1"XX2

"BB1"BB2

A:69Vectors "xx1 and "xx2 are given by

"XX1 "AA1 "AA2 "AA14

"AA3 1

"BB1 "AA2 "AA14

"BB2

A:70

"XX2 "AA14 "BB2 "AA3 "XX1

A:71

A.9 CROUTS TRANSFORMATION

A matrix can be resolved into the product of a lower triangular matrix "LL and an

upper unit triangular matrix "UU, i.e.,

a11 a12 a13 a14a21 a22 a23 a24a31 a32 a33 a34a41 a42 a43 a44

l11 0 0 0

l21 l22 0 0

l31 l32 l33 0

l41 l42 l43 l44

1 u12 u13 u140 1 u23 u240 0 1 u340 0 0 1

A:72

Matrix Methods 727



17/24

The elements of "UU and "LL can be found by multiplication:

l11 a11

l21 a21

l22 a22 l21u12

l31 a31l32 a32 l31u12

l33 a33 l31u13 l32u23

l41 a41

l42 a42 l41u12

l43 a43 l41u13 l42u23

l44 a44 a41u14 l42u24 l43u3

A:73

and

u12 a12=l11

u13 a13=l11

u14 a14=l11

u23 a23 l21u13=l22

u24 a24 l21u14=l22

u34 a34 l31u14 l32u24l33

A:74

In general:

lij aij Xkj1k1

likukj i ! j A:75

for j 1; . . . ; n

uij 1

liiaij

Xkj1k1

likukj

i < j A:76

Example A.10

Transform the following matrix into LU form:

1 2 1 0

0 3 3 1

2 0 2 0

1 0 0 2

From Eqs. (A.75) and (A.76):

1 2 1 0

0 3 3 1

2 0 2 0

1 0 0 2

1 0 0 0

0 3 0 0

2 4 4 0

1 2 1 2:33

1 2 1 0

0 1 1 0:33

0 0 1 0:33

0 0 0 1

728 Appendix A



18/24

The original matrix has been converted into a product of lower and upper triangular

matrices.

A.10 GAUSSIAN ELIMINATION

Gaussian elimination provides a natural means to determine the LU pair:

a11 a12 a13a21 a22 a23a31 a32 a33

x1x2x3

b1b2b3

A:77

First, form an augmented matrix:

a11 a12 a13 b1a21 a22 a23 b2a31 a32 a33 b3

A:78

1. Divide the first row by a11. This is the only operation to be carried out on

this row. Thus, the new row is

1 a 012 a013 b

01

a 012 a12=a11; a013 a13=a11; b

01 b1=a11

A:79

This gives

l11 a11; u11 1; u12 a012; u13 a

013 A:80

2. Multiply new row 1 by a21 and add to row 2. Thus, a21 becomes zero.

0 a 022 a023 a

033b

02

a 022 a22 a21a012

a 023 a23 a21a013

b 02 b2 a21b01

A:81

Divide new row 2 by a 022. Row 2 becomes

0 1 a 0023 b002

a 0023 a 023=a 022

b 002 b02=a

022

A:82

This gives

l21 a21; l22 a022; u22 1; u23 a

023 A:83

3. Multiply new row 1 by a31 and add to row 3. Thus, row 3 becomes:

0 a 032 a033b

03

a 032 a32 a32a012

a 033 a33 a31a013

A:84

Multiply row 2 by a32 and add to row 3. This row now becomes

0 0 a 0033 b003 A:85

Matrix Methods 729



19/24

Divide new row 3 by a 0033. This gives

0 0 1 b 0 003

b 0 003 b003 =a

0033

A:86

From these relations:

l33 a0033; l31 a31; l32 a

032; u33 1 A:87

Thus, all the elements of LU have been calculated and the process of forward

substitution has been implemented on vector "bb.

A.11 FORWARDBACKWARD SUBSTITUTION METHOD

The set of sparse linear equations:

"AA "xx "bb A:88

can be written as

"LL "UU"xx "bb A:89

or

"LL "yy "bb A:90

where

"yy "UU"xx A:91

"LL "yy "bb is solved for "yy by forward substitution. Thus, "yy is known. Then "UU"xx "yy is

solved by backward substitution.

Solve "LL "yy "bb by forward substitution:

l11 0 0 0

l21 l22 0 0

l31 l32 l33 0

l41 l42 l43 l44

y1y2y3y4

b1b2b3b4

A:92

Thus,

y1 b1=l11

y2 b2 l21y1=l22

y3 b3 l31y1 l32y2=l33

y4 b4 l41y1 l42y2 l43y3=l44

A:93

Now solve "UU"xx "yy by backward substitution:

1 u12 u13 u140 1 u23 u240 0 1 u340 0 0 1

x1x2x3x4

y1y2y3y4

A:94

730 Appendix A



20/24

Thus,

x4 y4

x3 y3 u34x4

x2 y2 u23x3 u24x4

x1 y1 u12x2 u13x3 u14x4

A:95

The forwardbackward solution is generalized by the following equation:

"AA "LL "UU "LLd "LLl "II "UUu A:96

where "LLd is the diagonal matrix, "LLl is the lower triangular matrix, "II is the identity

matrix, and "UUu is the upper triangular matrix.

Forward substitution becomes

"LL "yy "bb

"LLd "LLl "yy "bb

"LLd "yy "bb "LLl "yy

"yy "LL1d "bb "LLl "yy

A:97

i.e.,

y1y2y3

y4

1=l11 0 0 0

0 1=l22 0 0

0 0 1=l33 0

0 0 0 1=l44

x

b1b2b3

b4

0 0 0 0

l21 0 0 0

l31 l32 0 0

l41 l42 l43 l44

y1y2y3

y4

2

664

3

775A:98

Backward substitution becomes

"II "UUu "xx "yy

"xx "yy "UUu "xxA:99

i.e.,

x1x2x3x4

y1y2y3y4

0 u12 u13 u140 0 u23 u240 0 0 u340 0 0 0

x1x2x3x4

A:100

A.11.1 Bifactorization

A matrix can also be split into LU form by sequential operation on the columns and

rows. The general equations of the bifactorization method are

lip a1p for ! p

upj apj

appfor j > p

aij a1 j lipupj for i > p;j > p

A:101

Matrix Methods 731



21/24

Here, the letter p means the path or the pass. This will be illustrated with an

example.

Example A.11

Consider the matrix:

"AA

1 2 1 0

0 3 3 1

2 0 2 0

1 0 0 2

It is required to convert it into LU form. This is the same matrix of Example A.10.

Add an identity matrix, which will ultimately be converted into a U matrix and

the "AA matrix will be converted into an L matrix:

1 2 1 00 3 3 1

2 0 2 0

1 0 0 2

1 0 0 00 1 0 0

0 0 1 0

0 0 0 1

First step, p=1:

The shaded columns and rows are converted into L and U matrix column and row

and the elements of "AA matrix are modified using Eq. (A.101), i.e.,

a32 a32 l31u12

0 22 4

a33 a33 l31u13

2 21 0

Step 2, pivot column 2, p=2:

732 Appendix A

1 1 2 1 0

0 3 3 0 0 1 0 0

2 4 0 0 0 0 1 0

1 2 1 2 0 0 1 0

1 0 0 0 1 2 1 0

0 3 0 0 0 1 1 0.33

2 4 4 1.32 0 0 1

1 2 1 2.66 0 0 0 1



22/24

Third step, pivot column 3, p=3:

This is the same result as derived before in Example A.10.

A.12 LDU (PRODUCT FORM, CASCADE, OR CHOLESKI FORM)

The individual terms of L, D, and U can be found by direct multiplication. Again,

consider a 4 4 matrix:


1 0 0 0

l21 1 0 0

l31 l32 1 0

l41 l42 l43 1

d11 0 0 0

0 d22 0 0

0 0 d33 0

0 0 0 d44

1 u12 u13 u140 1 u23 u240 0 1 u340 0 0 1

A:102

The following relations exist:

d11 a11

d22 a22 l21d11u12

d33 a33 l31d11u13 l32d22u23

d44 a44 l41d11u14 l42d22u24 l43d33u34

u12 a12=d11

u13

a13

=d11

u14 a14=d11

u23 a23 l21d11u13=d22

u24 a24 l21d11u14=d22

u34 a34 l31d11u14 l32d22u24=d33

l21 a21=d11

l31 a31=d11

l32

a32

l31

d11

u12

=d22

l41 a41=d11

l42 a42 l41d11u12=d22

l43 a43 l41d11u13 l42d22u23=d33

A:103

Matrix Methods 733

1 0 0 0 1 2 1 0

0 3 0 0 0 1 1 0.33

2 4 4 0 0 0 1 0.33

1 2 1 2.33 0 0 0 1



23/24

734 Appendix A

In general:

dii a11 Xi1j1

lijdjjuji for i 1; 2; . . . ; n

uik aik Xi1

j1

lifdjjujk" #=dii for k i 1 . . . ; n i 1; 2; . . . ; nlki aki

Xi1j1

lkjdjjuji

" #=dii for k i 1; . . . ; n i 1; 2; . . . ; n

A:104

Another scheme is to consider A as a product of sequential lower and upper matrices

as follows:

A L1L2; . . . ; LnUn; . . . ; U2U1 A:105


l11 0 0 0

l21 1 0 0l31 0 1 0

l41 0 0 1

1 0 0 0

0 a222 a232 a2420 a322 a332 a3420 a422 a432 a442

1 u12 u13 u140 1 0 00 0 1 0

0 0 0 1

A:106

Here the second step elements are denoted by subscript 2 to the subscript.

l21 a21 l31 a31 l41 a41

u12 a12=l11 u13 a13=l11 u14 a14=l11

aij2

a1j

l1i

u1j

i;j 2; 3; 4

A:107

All elements correspond to step 1, unless indicated by subscript 2.

In general for the kth step:

dkkk akkk k 1; 2; . . . ; n 1

lkik akik=a

kkk

ukj akkj=a

kkk

ak1ij akij a

kika

kkj=a

kkk

k 1; 2; . . . ; n 1i;j k 1; . . . ; n

A:108

Example A.12

Convert the matrix of Example A.10 into LDU form:

1 2 1 0

0 3 3 1

2 0 2 0

1 0 0 2

l1 l2 l3 D u3 u2 u1

The lower matrices are

l1 l2 l3

1 0 0 0

0 1 0 0

2 0 1 0

1 0 0 1

1 0 0 0

0 1 0 0

0 4=3 1 0

1 2=3 0 1

1 0 0 0

0 1 0 0

0 0 1 0

0 0 1=4 0



24/24

The upper matrices are

u3 u2 u1

1 0 0 0

0 1 0 1=3

0 0 1 1=3

0 0 0 1

1 0 0 0

0 1 1 0

0 0 1 0

0 0 0 1

1 2 1 0

0 1 0 0

0 0 1 0

0 0 0 1

The matrix D is

D

1 0 0 0

0 3 0 0

0 0 4 0

0 0 0 7=3

Thus, the LDU form of the original matrix is

1 0 0 0

0 1 0 0

2 4=3 1 01 2=3 1=4 1

1 0 0 0

0 3 0 0

0 0 4 00 0 0 7=3

1 2 1 0

0 1 1 1=3

0 0 1 1=30 0 0 1

If the coefficient matrix is symmetrical (for a linear bilateral network), then

L Ut

A:109

Because

lipnew aip=app

upi api=appaip apiA:110

The LU and LDU forms are extensively used in power systems.

BIBLIOGRAPHY

1. PL Corbeiller. Matrix Analysis of Electrical Networks. Cambridge, MA: Harvard

University Press, 1950.

2. WE Lewis, DG Pryce. The Application of Matrix Theory to Electrical Engineering.

London: E&F N Spon, 1965.

3. HE Brown. Solution of Large Networks by Matrix Methods. New York: Wiley

Interscience, 1975.

4. SA Stignant. Matrix and Tensor Analysis in Electrical Network Theory. London:

Macdonald, 1964.

5. RB Shipley. Introduction to Matrices and Power Systems. New York: Wiley, 1976.

Matrix Methods 735

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