Massoud M. Engineering Thermofluids: Thermodynamics, Fluid Mechanics, and Heat Transfer

1. Definition of Terms 963

VIId. Linear Algebra

1. Definition of Terms

Row vector. A row vector is a group of real numbers written in a row. For ex-

ample, the row vector A is defined as

[]

n

aaaaA ,,,,

321

"= . Each individual

number in this vector is referred to as a component.

Column vector. A column vector is a group of real numbers written in a col-

umn. If the column vector has n numbers, then the column vector is an n-

dimensional vector or a vector of order n.

Equal vectors. Two row vectors or two column vectors are equal if they are of

the same order and have identical components.

Addition of vectors. Only vectors of the same type (row or column) and the

same order can be added. For example, the addition of two row vectors of order n

is given as:

[][][ ]

nnnn

bababababbbbaaaaBA ++++=+=+ ,,,,,,,,,,,,

332211321321

"""

Commutative laws of addition. For any n-dimensional row (or column) vec-

tors such as vectors A and B:

A + B = B + A

Associative laws of addition. If A, B, and C are any n-dimensional row (or

column) vectors:

(A + B) + C = A + (B + C)

Multiplication of a vector by a number. If A is an n-dimensional row vector,

then the product of A by a real number c is given as:

[]

n

cacacacacA ,,,,

321

"= .

Multiplication of vectors. If A is a row vector and B a column vector, then the

product of

B

A ⋅ is a scalar given by:

[]

()

»

¼

º

«

¬

ª

=++++=

»

¼

º

«

¬

ª

=⋅

¦

=

n

k

kknn

n

bababababa

b

aaaaBA

1

1111311321121111

1

31

21

11

1131211

,,,, "

#

"

Now consider multiplication of a column vector and a row vector:

964 VIId. Engineering Mathematics: Linear Algebra

[]

»

¼

º

«

¬

ª

=

»

¼

º

«

¬

ª

=⋅

11113112111

11311331123111

211211321122111

111111311121111

1131211

1

31

21

11

,,,,

nnnnn

nn

n

babababa

aaaa

b

BA

"

#####

"

#

The result is called a matrix as defined below.

The distributive law of vectors. If A is a row vector of dimension n and B and

C are column vectors of dimension n, the distributive law specifies that:

CABACBA ⋅+⋅=+⋅ )(

Matrix. A rectangular array of real numbers is called a matrix when arranged

as shown in Figure VIId.1.1(a). Note that in general, the number of rows and col-

umns are different.

¸

¹

·

¨

©

§

=

mnmm

n

aaa

A

"

####

"

21

22221

11211

¸

¹

·

¨

©

§

=

nnnn

n

aaa

A

"

####

"

21

22221

11211

(a) (b)

Figure VIId.1.1. Demonstrations of (a) a matrix and (b) a square matrix

Order of a matrix defines the number of rows and columns. For example, if a

matrix has

m rows and n columns then the matrix is of order m by n or alterna-

tively

m × n. Hence, a row vector is a 1 × m matrix and a column vector is an m

× 1 matrix. Two matrices of the same order are referred to as comforbale.

Square matrix. A

nm × matrix would be referred to as a square matrix if m

= n as shown in Figure VIIc.1.1(b).

Main diagonal. In a square matrix, the main diagonal consists of the a

ii

ele-

ments.

Zero matrix. All the elements of a zero matrix are zeros.

Upper triangular matrix. A square matrix with elements a

ij

= 0 for i > j is

called an upper triangular matrix (Figure VIId.1.2(a)).

Lower triangular matrix. A square matrix with elements a

ij

= 0 for i < j is

called a lower triangular matrix (Figure VIId.1.2(b)).

1. Definition of Terms 965

¸

¹

·

¨

©

§

=

nn

n

a

aa

aaa

A

"

####

"

00

0

222

11211

¸

¹

·

¨

©

§

=

nnnn

aaa

aa

a

A

"

####

"

21

2221

11

0

00

¸

¹

·

¨

©

§

=

nn

a

A

"

####

"

00

22

11

(a) (b) (c)

Figure VIId.1.2. Demonstration of (a) upper triangular, (b) lower triangular, and (c) di-

agonal matrix

Diagonal matrix. All elements of a diagonal matrix are zeroes except for the

elements of the main diagonal. In other words, a matrix which is both upper and

lower triangular, is called a diagonal matrix:

Scalar matrix. A scalar matrix is a diagonal matrix with elements such that

kaaa

nn

==== "

2211

.

Identity matrix. An identity matrix is a scalar matrix with k = 1.

Sums of matrices. If matrix

[

]

mn

ij

aA = is added to (or deducted from) matrix

[

]

mn

ij

bB = , the result would be matrix C with elements given as

[

]

mn

ijij

baC ±= . The commutative and associative laws of addition described for

vectors also apply to matrices. Only matrices of the same order can be summed.

Multiplication of matrices. The product of matrix

[

]

mn

ij

aA = by matrix

[

]

np

ij

bB = is matrix

[

]

mp

ij

cC = with elements given as:

¸

¹

·

¨

©

§

mnmm

inii

n

aaa

"

#"##

"

#"##

"

21

11211

¸

¹

·

¨

©

§

npnjn

pj

bbb

""

#"#"#

""

1

2221

1111

=

¸

¹

·

¨

©

§

mpmjm

ip

j

ii

pj

ccc

#"

###

""

###

""

1

1111

Where element c

ij

of matrix C is given by:

njinjijiji

bababac +++= "

2211,

As shown above, in multiplication of two matrices A and B, these need not be

square matrices. However, the number of columns of matrix A must be equal to

the number of rows of matrix B. It is also important to note that matrix multiplica-

tion is not a linear operation hence,

.BAAB ≠ An example of multiplication of

two square matrices is as follows

¸

¹

·

¨

©

§

−

=

¸

¹

·

¨

©

§

−

−−

−

¸

¹

·

¨

©

§

−

114916

31566

151020

816

531

6117

315

493

211

966 VIId. Engineering Mathematics: Linear Algebra

An example of the multiplication of two non-square matrices, A(n

a

, m)B(m, m

b

) =

C(n

a

, m

b

), is:

»

¼

º

«

¬

ª

−

=

»

¼

º

«

¬

ª

−

»

¼

º

«

¬

ª

−−

−

3755

16342

11574

129

63

72

40

51

49126

75370

1011851

Transpose of a matrix. The transpose of matrix A is matrix A

T

, obtained by

interchanging the rows and columns of A:

¸

¹

·

¨

©

§

=

mnmm

n

aaa

A

"

####

"

21

22221

11211

,

¸

¹

·

¨

©

§

=

mnnn

m

T

aaa

A

"

####

"

21

22212

12111

Note that (A

T

)

T

= A, (A + B)

T

= A

T

+ B

T

, (AB)

T

= B

T

A

T

. Furthermore, matrix A is

said to be symmetric if A = A

T

and is said to be skew symmetric if A = – A

T

.

Determinant of a square matrix. A determinant of a square matrix is a func-

tional value assigned to the numbers in the square array. While a matrix is repre-

sented by an array of numbers within brackets, the determinant of the matrix is

represented by the same array of numbers within two parallel lines:

¸

¹

·

¨

©

§

=

nnnn

n

aaa

A

"

####

"

21

22221

11211

nnnn

n

aaa

A

"

####

"

21

22221

11211

=

Minor determinant of a square matrix. If we eliminate the row and column

associated with element a

ij

of matrix A then the remaining array of numbers make

up the elements of a minor determinant shown by M

ij

. Therefore, there are as

many minor determinants as the number of elements in the matrix. Note that if the

order of matrix A is n, the order of each minor determinant is n – 1. . For exam-

ple, for element a

12

, of matrix A, the minor determinant is found as:

nnnn

n

aaa

A

"

####

"

21

22221

11211

= ,

nnn

n

aa

M

"

###

"

1

221

=

Example: Let’s find the minor determinant corresponding to elements a

22

, a

23

,

and a

32

of matrix A below:

1. Definition of Terms 967

¸

¹

·

¨

©

§

−

=

843

315

261

A ,

83

21

22

−

=M

,

43

61

23

−

=M

,

35

21

32

−

=M

,

Cofactor of an element of a matrix. The minor determinant multiplied by

(–1)

i+j

is called the cofactor of element a

ij

of matrix A and is designated as

c

a

ij

A .

For example, the cofactors of a

11

, a

23

, and a

33

of matrix A below are obtained as:

¸

¹

·

¨

©

§

−

−−

=

279

450

312

A

,

27

45

)1(

2

11

−

−=

C

A

,

79

12

)1(

5

23

−

−−

−=

C

A

,

50

12

)1(

6

33

−−

−=

C

A

The cofactor matrix. A matrix whose elements are the cofactors of the ele-

ments of matrix A is known as the cofactor matrix of A. A matrix and its associ-

ated cofactor matrix are written as:

¸

¹

·

¨

©

§

=

nnnn

n

aaa

A

"

####

"

21

22221

11211

,

¸

¹

·

¨

©

§

=

c

nn

c

n

c

n

c

n

cc

c

n

cc

c

AAA

A

"

####

"

21

22221

11211

Example: Consider matrix A and its associated cofactor matrix (A

c

):

¸

¹

·

¨

©

§

−

−−

=

679

851

325

A

¸

¹

·

¨

©

§

−−

−−−

=

23371

53579

527826

c

A

Determinant of a square matrix. For matrix A of order n, the value of the de-

terminant is given as:

¦¦

=−=

+ c

a

ijijij

ji

ij

AaMaA )1( VIId.1.1

where

ij

jic

a

MA

ij

+

−= )1( is the cofactor and M

ij

is the minor determinant associ-

ated with element a

ij

. This minor determinant is of order n – 1 and obtained by

deleting the ith row and jth column, as discussed before. For a matrix of order n,

there are n! terms to evaluate. Consider first a matrix of order 2. There are 2! = 2

terms to evaluate. Using Equation VIId.1.1, the determination of a matrix of or-

der 2 becomes:

21122211

2221

1211

aaaa

aa

A −==

968 VIId. Engineering Mathematics: Linear Algebra

We now calculate the determinant of a matrix of order 3. In this case, there are 3!

= 6 terms to evaluate:

)()()(

221323123132133312212332332211

2322

1312

31

3332

1312

21

3332

2322

11

333231

232221

131211

aaaaaaaaaaaaaaa

aa

a

aa

a

aa

a

aaa

A

−+−−−

=+−==

Example: The determinant of matrices A, B, and C are calculated as follows:

721

1310

561

942

−=

−−

−

=A

,

4

774340

512827

271514

=

−

−−

−

=B

,

1000

305045

403015

252010

=

−

−−

−

=C

Singular matrix. If the determinant of a matrix is zero, then the matrix is sin-

gular.

Adjoint matrix. The transpose of the cofactor matrix is called the adjoint ma-

trix. Matrix A, the cofactor matrix A

c

, and the adjoint matrix of A are shown in

Figure VIId.1.3(a), (b), and (c), respectively.

¸

¹

·

¨

©

§

=

nnnn

n

aaa

A

"

####

"

21

22221

11211

,

¸

¹

·

¨

©

§

=

nnnn

n

AAA

C

"

####

"

21

22221

11211

,

¸

¹

·

¨

©

§

=

nnnn

n

AAA

adjA

"

####

"

21

22212

12111

(a) (b) (c)

Figure VIId.1.3. Demonstration of (a) square matrix, (b) the cofactor matrix, and (c) the

adjoint matrix.

Diagonally dominant matrix. In diagonally dominant matrices, the magnitude

of the element located on the diagonal in each row is larger than the sum of the

magnitude of all the other elements in that row:

¦

≠

>

ijall

ijii

aa

2. The Inverse of a Matrix

The inversion of a matrix is very important in determining solutions to sets of al-

gebraic equations. The process of matrix inversion is analogous to division for

real numbers. If matrix B is the inverse of matrix A, then the product of matrices

A and B is equal to the identity matrix, AB = I. The inverse of a matrix is denoted

2. The Inverse of a Matrix 969

¸

¹

·

¨

©

§

−−

−

=

4111

792

1105

A

,

¸

¹

·

¨

©

§

−−−

=

253779

115941

1018529

c

A

, C =

()

¸

¹

·

¨

©

§

−−

==

25115101

37985

794129

adjAA

T

c

(a) (b) (c)

Figure VIId.2.1. Demonstration of (a) a square matrix, (b) the cofactor matrix, and (c)

the adjoint matrix

as B = A

– 1

. But not every matrix has an inverse. There are two conditions for a

matrix to have an inverse. The first condition requires the matrix to be square so

that a determinant can be calculated. The second condition requires the determi-

nant of the matrix to be nonzero otherwise the matrix is singular. To inverse ma-

trix A, we first replace all elements by their associated cofactors to obtain the co-

factor matrix, A

c

. We then transpose the cofactor matrix to obtain the adjoint

matrix. The inverse of the matrix is the result of the division of the elements of

the adjoint matrix by the determinant of matrix A:

A

adjA

A =

−1

For example, if we want to find the inverse of matrix A, Figure VIId.2.1(a), we

must first find the cofactor matrix by replacing each element of matrix A by its

minor determinant, using the correct sign. This is shown in Figure VIId.2.1(b).

The adjoint matrix of A is then obtained by transposing matrix A

c

. Finally, the in-

verse matrix A

–1

is obtained by dividing elements of matrix C by the determinant

of matrix A.

Elements of the cofactor matrix are obtained as follows. For element a

11

= –5,

we substitute from the minor determinant obtained from eliminating elements in

the first row and the first column of matrix A to get [

)71(49 ×−−×− ] = – 29. We

retain the sign as the summation of first row and first column is an even number.

Similarly, for element a

21

= 2 we substitute from )]1(1410[ −×−× = 41. How-

ever, the sign for this term would be minus since the summation of the row and

column is an odd number.

We now calculate the determinant of matrix A. This is facilitated as we already

have the minor matrices:

806)79(11)41(2)29(5

4111

792

1105

−=−−−−=

−−

−

=A

Having the adjoint matrix, Figure VIId.2.1(c) and the determinant, the inverse ma-

trix becomes:

970 VIId. Engineering Mathematics: Linear Algebra

¸

¹

·

¨

©

§

−

=

−

0310.01430.0125.0

0459.00112.0105.0

0980.00509.0036.0

1

A

To ensure that we have made no algebraic error, we should calculate the product

of AA

-1

= I:

¸

¹

·

¨

©

§

−−

−

4111

792

1105

¸

¹

·

¨

©

§

−

×

0310.01430.0125.0

0459.00112.0105.0

0980.00509.0036.0

=

¸

¹

·

¨

©

§

100

010

001

Several matrices and their inverse matrices are provided below. The reader should

try to follow the above procedure to inverse each matrix and compare the results

with the inverse matrices presented.

Example 1:

¸

¹

·

¨

©

§

−

−−=

1271

409

213

A ,

¸

¹

·

¨

©

§

−

−−−

=

−

06160.01510.04320.0

0411.02330.07120.0

0274.0178.0192.0

1

A , 146=A

Example 2:

¸

¹

·

¨

©

§

−−

−

−−

=

630.075.025.0

4000.056.030.0

079.000.115.0

A

,

¸

¹

·

¨

©

§

−−−

−

=

−

46.257.1968.0

953.030.101.1

060.585.7601.0

1

A

, 0878.0−=A

Example 3:

1

10 14 15 8 0.393 0.194 0.113 0.188

13 3 21 13 0.0441 .0089 0.0229 0.072

,

18 12 7 11 1.12 0.431 0.243 0.602

2 19 14 10 1.40 0.581 0.274 0.769

AA

−

−− −

§·§ ·

¨¸¨ ¸

−− −−

¨¸¨ ¸

==

¨¸¨ ¸

−− − − −

¨¸¨ ¸

−− −−−

©¹© ¹

, 6980=A

Example 4:

¸

¹

·

¨

©

§

−

−−

=

9023

10108

11932

7654

A

,

¸

¹

·

¨

©

§

−−

−

−−−

=

−

091.0168.0160.0269.

106.0234.0977.0162.

101.0762.1210.1630.

127.1010.0323.0315.

1

A

, 5550A =−

Such operations can be easily performed by using the accompanying CD-ROM.

3. Set of Linear Equations 971

3. Set of Linear Equations

Mathematical modeling of most physical phenomena reduces to a set of simulta-

neous differential equations the solution of which would determine the parameters

of interest. The solution to such set of equations involves the approximation of

the differential terms by finite difference, for example, and then linearization of

the nonlinear terms. The linearization of a set of non-linear differential equations

is discussed in Chapter VIIe. The net result is a set of linear simultaneous equa-

tions as given in Equation VIId.3.1, which must be solved in each time interval to

obtain the trend of the parameters. There are several techniques for solving a set

of n simultaneous linear equations in n unknowns:

°

¯

°

®



=++++

nnnnnnn

nn

cxaxaxaxa

"

#

"

332211

22323222121

11313212111

VIId.3.1

such as matrix inversion and the Gauss – Seidel iteration method.

3.1. Solution to a Set of Linear Equations by Matrix Inversion

The above general set of linear equations can be written in the form of a matrix

equation as:

Ax = c VIId.3.2

where A is the coefficient matrix, x is the unknown vector, and c is the constant

vector. If we now multiply both sides of Equation VIId.3.2 by A

–1

we get A

–1

(Ax)

= A

–1

c. Since A

–1

A = I, the left hand-side becomes:

x = A

–1

c

As a result, to solve a set of algebraic equations, we must find the inverse of the

coefficient matrix and multiply it by the vector of the constants. If we place the

vector of constants c as the last column inside the coefficient matrix A, the resul-

tant is called the augmented matrix. Several examples are provided below. The

reader should try to solve these sets and compare the results with those given be-

low.

Example 1. Consider the set of linear equations as shown in Figure VIId.3.1(a).

This set in matrix form is shown in Figure VIId.3.1(b). The augmented matrix of

this set is shown is Figure VIId.3.1(c).

972 VIId. Engineering Mathematics: Linear Algebra

°

¯

°

®



−=−+−

−=+

=+++−

=−+−

106537

76

1394

5232

4321

41

4321

xxxx

xx

xxxx

,

¸

¹

·

¨

©

§

−

=

¸

¹

·

¨

©

§

¸

¹

·

¨

©

§

−−

−

−−

10

7

1

5

6537

6001

3941

2132

4

3

2

1

x

,

¸

¹

·

¨

©

§

−−−

−

−−

106537

76001

13941

52132

(a) (b) (c)

Figure VIId.3.1. Demonstration of (a) set of linear equations, (b) matrix form, and (c)

augmented matrix

Upon solving the above set, the solution vector becomes x = [– 4.52, –3.91, 1.48, –

0.414]

T

.

Example 2:

°

¯

°

®



=−−+

−=++−

−=−−+

=+−−

1034137

566

1013344

15212

zyxw

,

¸

¹

·

¨

©

§

−

=

¸

¹

·

¨

©

§

¸

¹

·

¨

©

§

−−

−

−−

10

5

10

15

34137

6611

13344

21112

z

y

x

w

,

¸

¹

·

¨

©

§

−−

−−−

−−

1034137

56611

1013344

1521112

The answers obtained for x

1

through x

4

is arranged in the transpose of the solution

vector:

x

T

= [0.0343, 0.405, – 2.18, 1.41]

T

.

3.2. Solution to a Set of Linear Equations by Gauss – Siedel Iteration

This is a simple and effective method to solve sets of linear algebraic equations.

To explain the Gauss – Seidel iteration, let’s consider a set of 3 equations in 3 un-

knowns:

°

¯

°

®



=++

3333232131

2323222121

1313212111

cxaxaxa

VIId.3.3

This set can be arranged as:

¸

¹

·

¨

©

§

=

¸

¹

·

¨

©

§

¸

¹

·

¨

©

§

3

2

1

3

2

1

333231

232221

131211

c

x

aaa

We have arranged this set so that the diagonal elements in the coefficient matrix

have the largest magnitude in each row. For example, for row 2 we have a

22

> a

21

and a

22

> a

23

, etc. This arrangement expedites the iteration process. In fact, if the

diagonal elements are larger than the summation of other elements of their corre-

Massoud M. Engineering Thermofluids: Thermodynamics, Fluid Mechanics, and Heat Transfer

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