Matrix algebra.PDF

Matrix Algebra

and Solution

of Matrix Equations

1.1 INTRODUCTION

notation and

their manipulations and applications. Vector is a one-dimensional array of numbers

and/or characters arranged as a single column. The number of rows is called the

matrix

and

vector

of that vector. Matrix is an extension of vector when a set of numbers and/or

characters are arranged in rectangular form. If it has M rows and N column, this

matrix then is said to be of order M by N. When M = N, then we say this

square

matrix is of order N (or M). It is obvious that vector is a special case of matrix when

there is only one column. Consequently, a vector is referred to as a column matrix

as opposed to the row matrix which has only one row. Braces are conventionally

used to indicate a vector such as {V} and brackets are for a matrix such as [M].

In writing a computer program, DIMENSION or DIM statements are necessary

to declare that a certain variable is a vector or a matrix. Such statements instruct

the computer to assign multiple memory spaces for keeping the values of that vector

or matrix. When we deal with a large number of different entities in a group, it is

better to arrange these entities in vector or matrix form and refer to a particular

entity by specifying where it is located in that group by pointing to the row (and

column) number(s). Such as in the case of having 100 numbers represented by the

variable names A, B, …, or by A(1) through A(100), the former requires 100 different

characters or combinations of characters and the latter certainly has the advantage

of having only one name. The A(1) through A(100) arrangement is to adopt a vector;

these numbers can also be arranged in a matrix of 10 rows and 10 columns, or 20

rows and ﬁve columns depending on the characteristics of these numbers. In the

cases of collecting the engineering data from tests of 20 samples during ﬁve different

days, then arranging these 100 data into a matrix of 20 rows and ﬁve columns will

be better than of 10 rows and 10 columns because each column contains the data

collected during a particular day.

In the ensuing sections, we shall introduce more deﬁnitions related to vector

and matrix such as transpose, inverse, and determinant, and discuss their manipula-

tions such as addition, subtraction, and multiplication, leading to the organizing of

systems of linear algebraic equations into matrix equations and to the methods of

ﬁnding their solutions, speciﬁcally the Gaussian Elimination method. An apparent

application of the matrix equation is the transformation of the coordinate axes by a

Computers are best suited for repetitive calculations and for organizing data into

specialized forms. In this chapter, we review the

order

rotation about any one of the three axes. It leads to the derivation of the three basic

transformation matrices and will be elaborated in detail.

Since the interactive operations of modern personal computers are emphasized

in this textbook, how a simple three-dimensional brick can be displayed will be

discussed. As an extended application of the display monitor, the transformation of

coordinate axes will be applied to demonstrate how animation can be designed to

simulate the continuous rotation of the three-dimensional brick. In fact, any three-

dimensional object could be selected and its motion animated on a display screen.

Programming languages,

FORTRAN

QuickBASIC

MATLAB

, and

Mathe-

are to be initiated in this chapter and continuously expanded into higher

levels of sophistication in the later chapters to guide the readers into building a

collection of their own programs while learning the computational methods for

solving engineering problems.

1.2 MANIPULATION OF MATRICES

Two matrices [A] and [B] can be added or subtracted if they are of same order, say

M by N which means both having M rows and N columns. If the sum and difference

matrices are denoted as [S] and [D], respectively, and they are related to [A] and

[B] by the formulas [S] = [A] + [B] and [D] = [A]-[B], and if we denote the elements

in [A], [B], [D], and [S] as a

, b

, d

, and s

sab

(1)

and

dab

=−

(2)

Equations 1 and 2 indicate that the element in the ith row and jth column of [S]

is the sum of the elements at the same location in [A] and [B], and the one in [D]

is to be calculated by subtracting the one in [B] from that in [A] at the same location.

To obtain all elements in the sum matrix [S] and the difference matrix [D], the index

i runs from 1 to M and the index j runs from 1 to N.

In the case of

vector

} is directed

from the origin of the x-y coordinate axes, point O, to the point 1 on the x-axis

which has coordinates (x

) = (4,0) and the other vector {V

} is directed from the

origin O to the point 2 on the y-axis which has coordinates (x

) = (0,3). One may

} which is the vector directed from

the origin to the point 3 whose coordinates are (x

} + {V

) = (4,3), or, one may want to

} which is the vector directed from the

origin O to the point 4 whose coordinates are (x

} – {V

) = (4,–3). In fact, the vector

}

which is a vector directed from the origin O to the point 5 whose coordinates are

} to the negative image of {V

}, namely {V

2–

). Mathematically, based on Equations 1 and 2, we can have:

matica

for i = 1 to M and j = 1 to N, respectively,

then the elements in [S] and [D] are to be calculated with the equations:

addition and subtraction, only one column is involved (N =

1). As an example of addition and subtraction of two vectors, consider the two

vectors in a two-dimensional space as shown in Figure 1 , one vector {V

want to ﬁnd the resultant of {R} = {V

ﬁnd the difference vector {D} = {V

{D} can be obtained by adding {V

{} = {} + {} =

and

{} = {} − {} =

−

} but are not on either one of the coordinate axes, such

as {D} and {E} in Figure 1 , we then have the sum vector {F} = {D} + {E} which

has components of 1 and –2 units along the x- and y-directions, respectively. Notice

that O467 forms a parallelogram in Figure 1 and the two vectors {D} and {E} are

the two adjacent sides of the parallelogram at O. To ﬁnd the sum vector {F} of {D}

and {E} graphically, we simply draw a diagonal line from O to the opposite vertex

of the parallelogram — this is the well-known

}, {V

}, and {V

2–

It should be evident that to write out a vector which has a large number of rows

will take up a lot of space. If this vector can be rotated to become from one column

to one row, space saving would then be possible. This process is called transposition

as we will be leading to it by ﬁrst introducing the length of a vector.

For the calculation of the

Law of Parallelogram

length

of a two-dimensional or three-dimensional vector,

} in Figure 1 , it would be a simple matter because they are

oriented along the directions of the coordinate axes. But for the vectors such as {R}

} and {V

FIGURE 1.

Two vectors in a two-dimensional space.

RV V

DV V

−

When Equation 1 is applied to two arbitrary two-dimensional vectors which

unlike {V

such as {V

and {D} shown in Figure 1 , the calculation of their lengths would need to know the

components

of these vectors in the coordinate axes and then apply the

Pythagorean

= 3 units along

the x- and y-axis, respectively, its length, here denoted with the symbol

. Since the vector {R} has components equal to r

= 4 and r

, is:

{} =+

[ ]

2 2 05

43 5

(3)

, its length is to be calculated with the

following formula obtained from extending Equation 3 from two-dimensions to N-

dimensions:

through v

{} =++…+

[

]

(4)

For example, a three-dimensional vector has components v

= v

= 4, v

= v

= 13.

We shall next show that Equation 4 can also be derived through the introduction of

the multiplication rule and transposition of matrices.

= v

= 12, then the length of this vector is

{V}

= [4

+ 3

+ 12

]

0.5

1.2 MULTIPLICATION OF MATRICES

A matrix [A] of order L (rows) by M (columns) and a matrix [B] of order M

by N can be multiplied in the order of [A][B] to produce a new matrix [P] of order

L by N. [A][B] is said as [A]

post-multiplied

by [B], or, [B]

pre-multiplied

by [A].

The elements in [P] denoted as p

for i = 1 to N and j = 1 to M are to be calculated

by the formula:

∑

a b

(5)

ik kj

in the ith row and jth column

of the product matrix [P] is to be calculated by multiplying the elements in the ith

row of the matrix [A] by the corresponding elements in the jth column of the matrix

[B]. It is therefore evident that the number of elements in the ith row of [A] should

be equal to the number of elements in the jth column of [B]. In other words, to

apply Equation 5 for producing a product matrix [P] by multiplying a matrix [A]

on the right by a matrix [B] (or, to say multiplying a matrix [B] on the left by a

matrix [A]), the number of columns of [A] should be equal to the number of row

of [B]. A matrix [A] of order L by M can therefore be post-multiplied by a matrix

[B] of order M by N; but [A] cannot be pre-multiplied by [B] unless L is equal to N!

As a numerical example, consider the case of a square, 3

3 matrix post-

multiplied by a rectangular matrix of order 3 by 2. Since L = 3, M = 3, and N = 2,

the product matrix is thus of order 3 by 2.

theorem

Rr r

To facilitate the calculation of the length of a generalized vector {V} which has

N components, denoted as v

Vvv N

3, and v

Equation 5 indicates that the value of the element p

123

456

789

−

16 2 5 34

46 55 64

76 85 94

() +

()

13 22 31

43 52 61

− () + () + ()

() +

()

() +

()

− () + () + ()

38291

61012

++ −−−

343

−

24 25 24

12 10 5

42 40 32

21 16 9

114

More exercises are given in the Problems listed at the end of this chapter for

the readers to practice on the matrix multiplications based on Equation 5.

It is of interest to note that the square of the length of a vector {V} which has

N components as deﬁned in Equation 4,

{V}

which is a row matrix of order

1 by N (one row and N columns). That is:

{} = {}{} =++…+

(6)

where the row index i ranges from 1

to L and the column index j ranges from 1 to M, the transpose of this matrix when

its elements are designated as t

For a L-by-M matrix having elements e

where the row index

r ranges from 1 to M and the column index c ranges from 1 to M because this

transpose matrix is of order M by L. As a numerical example, here is a pair of a

will have a value equal to e

2 matrix [G] and its 2

3 transpose [H]:

−

6 5 4

321

[] =

and

[] = [] =

−−−

If the elements of [G] and [H] are designated respectively as g

and h

, then

= –1, and

so on. There will be more examples of applications of Equations 5 and 6 in the

ensuing sections and chapters.

Having introduced the transpose of a matrix, we can now conveniently revisit

the addition of {D} and {E} in Figure 1 in algebraic form as {F} = {D} + {E} =

[4 –3]

= g

. For example, from above, we observe that h

= g

= 5, h

= g

. The resulting sum vector is indeed

correct as it is graphically veriﬁed in Figure 1 . The saving of space by use of

transposes of vectors (row matrices) is not evident in this case because all vectors

are two-dimensional; imagine if the vectors are of much higher order.

Another noteworthy application of matrix multiplication and transposition is to

reduce a system of linear algebraic equations into a simple, (or, should we say a

single)

+ [–3 1]

= [4+(–3) –3+1]

= [1 –2]

. For example, if we have three unknowns x, y, and z which

are to be solved from the following three linear algebraic equations:

matrix equation

() +

, can be obtained by application of

Equation 5 to {V} and its transpose denoted as {V}

VVVv v

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