CH-01 (Pao - Engineering Analysis)

1Matrix Algebraand Solutionof Matrix Equations1.1 INTRODUCTIONComputers are best suited for repetitive calculations and for organizing data intospecialized forms. In this chapter, we review the matrix and vector notation andtheir manipulations and applications. Vector is a one-dimensional array of numbersand/or characters arranged as a single column. The number of rows is called theorder of that vector. Matrix is an extension of vector when a set of numbers and/orcharacters are arranged in rectangular form. If it has M rows and N column, thismatrix then is said to be of order M by N.

When M = N, then we say this squarematrix is of order N (or M). It is obvious that vector is a special case of matrix whenthere is only one column. Consequently, a vector is referred to as a column matrixas opposed to the row matrix which has only one row. Braces are conventionallyused 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 necessaryto declare that a certain variable is a vector or a matrix. Such statements instructthe computer to assign multiple memory spaces for keeping the values of that vectoror matrix.

When we deal with a large number of different entities in a group, it isbetter to arrange these entities in vector or matrix form and refer to a particularentity by specifying where it is located in that group by pointing to the row (andcolumn) number(s).

Such as in the case of having 100 numbers represented by thevariable names A, B, …, or by A(1) through A(100), the former requires 100 differentcharacters or combinations of characters and the latter certainly has the advantageof 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 20rows and five columns depending on the characteristics of these numbers. In thecases of collecting the engineering data from tests of 20 samples during five differentdays, then arranging these 100 data into a matrix of 20 rows and five columns willbe better than of 10 rows and 10 columns because each column contains the datacollected during a particular day.In the ensuing sections, we shall introduce more definitions related to vectorand matrix such as transpose, inverse, and determinant, and discuss their manipulations such as addition, subtraction, and multiplication, leading to the organizing ofsystems of linear algebraic equations into matrix equations and to the methods offinding their solutions, specifically the Gaussian Elimination method.

An apparentapplication of the matrix equation is the transformation of the coordinate axes by a© 2001 by CRC Press LLCrotation about any one of the three axes. It leads to the derivation of the three basictransformation matrices and will be elaborated in detail.Since the interactive operations of modern personal computers are emphasizedin this textbook, how a simple three-dimensional brick can be displayed will bediscussed. As an extended application of the display monitor, the transformation ofcoordinate axes will be applied to demonstrate how animation can be designed tosimulate the continuous rotation of the three-dimensional brick.

In fact, any threedimensional object could be selected and its motion animated on a display screen.Programming languages, FORTRAN, QuickBASIC, MATLAB, and Mathematica are to be initiated in this chapter and continuously expanded into higherlevels of sophistication in the later chapters to guide the readers into building acollection of their own programs while learning the computational methods forsolving engineering problems.1.2 MANIPULATION OF MATRICESTwo matrices [A] and [B] can be added or subtracted if they are of same order, sayM by N which means both having M rows and N columns. If the sum and differencematrices 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 elementsin [A], [B], [D], and [S] as aij, bij, dij, and sij 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:sij = a ij + b ij(1)d ij = a ij − b ij(2)andEquations 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 indexi runs from 1 to M and the index j runs from 1 to N.In the case of vector addition and subtraction, only one column is involved (N =1).

As an example of addition and subtraction of two vectors, consider the twovectors in a two-dimensional space as shown in Figure 1, one vector {V1} is directedfrom the origin of the x-y coordinate axes, point O, to the point 1 on the x-axiswhich has coordinates (x1,y1) = (4,0) and the other vector {V2} is directed from theorigin O to the point 2 on the y-axis which has coordinates (x2,y2) = (0,3). One maywant to find the resultant of {R} = {V1} + {V2} which is the vector directed fromthe origin to the point 3 whose coordinates are (x3,y3) = (4,3), or, one may want tofind the difference vector {D} = {V1} – {V2} which is the vector directed from theorigin O to the point 4 whose coordinates are (x4,y4) = (4,–3). In fact, the vector{D} can be obtained by adding {V1} to the negative image of {V2}, namely {V2–}which is a vector directed from the origin O to the point 5 whose coordinates are(x5,y5). Mathematically, based on Equations 1 and 2, we can have:© 2001 by CRC Press LLC 4  0 4 4  0 4{R} = {V1} + {V2 } =   +   =  0   3  3 and{D} = {V1} − {V2 } =   −   =  0  3 −3When Equation 1 is applied to two arbitrary two-dimensional vectors whichunlike {V1}, {V2}, and {V2–} but are not on either one of the coordinate axes, suchas {D} and {E} in Figure 1, we then have the sum vector {F} = {D} + {E} whichhas components of 1 and –2 units along the x- and y-directions, respectively.

Noticethat O467 forms a parallelogram in Figure 1 and the two vectors {D} and {E} arethe 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 vertexof the parallelogram — this is the well-known Law of Parallelogram.It should be evident that to write out a vector which has a large number of rowswill take up a lot of space.

If this vector can be rotated to become from one columnto one row, space saving would then be possible. This process is called transpositionas we will be leading to it by first introducing the length of a vector.For the calculation of the length of a two-dimensional or three-dimensional vector,such as {V1} and {V2} in Figure 1, it would be a simple matter because they areoriented along the directions of the coordinate axes. But for the vectors such as {R}FIGURE 1. Two vectors in a two-dimensional space.© 2001 by CRC Press LLCand {D} shown in Figure 1, the calculation of their lengths would need to know thecomponents of these vectors in the coordinate axes and then apply the Pythagoreantheorem.

Since the vector {R} has components equal to rx = 4 and ry = 3 units alongthe x- and y-axis, respectively, its length, here denoted with the symbol , is:{R} = [rx2 + ry2 ] = [42 + 32 ] = 50.50.5(3)To facilitate the calculation of the length of a generalized vector {V} which hasN components, denoted as v1 through vN, its length is to be calculated with thefollowing formula obtained from extending Equation 3 from two-dimensions to Ndimensions:{V} = [v12 + v22 + … + v2N ]0.5(4)For example, a three-dimensional vector has components v1 = vx = 4, v2 = vy =3, and v3 = vz = 12, then the length of this vector is {V} = [42 + 32 + 122]0.5 = 13.We shall next show that Equation 4 can also be derived through the introduction ofthe multiplication rule and transposition of matrices.1.2 MULTIPLICATION OF MATRICESA matrix [A] of order L (rows) by M (columns) and a matrix [B] of order Mby N can be multiplied in the order of [A][B] to produce a new matrix [P] of orderL by N.

[A][B] is said as [A] post-multiplied by [B], or, [B] pre-multiplied by [A].The elements in [P] denoted as pij for i = 1 to N and j = 1 to M are to be calculatedby the formula:p ij =∑Mk =1a ik b kj(5)Equation 5 indicates that the value of the element pij in the ith row and jth columnof the product matrix [P] is to be calculated by multiplying the elements in the ithrow 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] shouldbe equal to the number of elements in the jth column of [B].

In other words, toapply 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 amatrix [A]), the number of columns of [A] should be equal to the number of rowof [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 postmultiplied 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.© 2001 by CRC Press LLC14 73  66 59 4258−3 1(6) + 2(5) + 3( 4) 1( −3) + 2( −2) + 3( −1) −2  = 4(6) + 5(5) + 6( 4) 4( −3) + 5( −2) + 6( −1)−1  7(6) + 8(5) + 9( 4) 7( −3) + 8( −2) + 9( −1) 6 + 10 + 12=  24 + 25 + 2442 + 40 + 32−3 − 4 − 3   28−12 − 10 − 5 =  73−21 − 16 − 9  114−10 −27−46More exercises are given in the Problems listed at the end of this chapter forthe 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 hasN components as defined in Equation 4, {V}2, can be obtained by application ofEquation 5 to {V} and its transpose denoted as {V}T which is a row matrix of order1 by N (one row and N columns).