Publisher Summary

The approach in this chapter is based on technique rather than on rigorous mathematical theories. It commences with various matrix definitions, followed by the laws of matrix algebra. To demonstrate the latter, several examples are worked out in detail and particular attention is paid to the inverse of a matrix and the solution of homogeneous and non-homogeneous simultaneous equations. A vector has both magnitude and direction, and typical vector quantities appear in the form of velocity, displacement, force, weight, etc. A matrix in its most usual forms is an array (or table) of scalar quantities, consisting of m rows by n columns. The elements of the matrix need not necessarily be scalars, but can take the form of vectors or even matrices. This compact method of representing quantities allows matrices to be particularly suitable for modeling physical problems on digital computers. A null matrix is one that has all its elements equal to zero. A diagonal matrix is a square matrix where all the elements except those of the main diagonal are zero.

1.1 DEFINITIONS

A scalar in its most usual form can be described as a number which is positive or negative or zero. Typical examples of scalars are 1, 2, π, e, −1.57, 2 × 1011, etc., and typical scalar quantities appear in the form of temperature, time, mass, length, etc. Scalars have only magnitude.

A vector has both magnitude and direction, and typical vector quantities appear in the form of velocity, displacement, force, weight, etc.

A matrix in its most usual forms is an array (or table) of scalar quantities, consisting of m rows by n columns, as shown in (1.1). The elements of the matrix need not necessarily be scalars, but can take the form of vectors or even matrices. This compact method of representing quantities allows matrices to be particularly suitable for modelling physical problems on digital computers.

A]=[A11A12A13.....A1nA21A22A23  .....A2nA31A32A33.....A3n............Am1Am2Am3.....Amn]. (1.1)

A row of a matrix is defined as a horizontal line of quantities.

A column of a matrix is defined as a vertical line of quantities.

The quantities A11, A12, A13, etc. are said to be the elements of the matrix [A].

The order of a matrix is defined by its number of rows x its number of columns. Thus, the matrix of (1.1) is said to be of order m x n.

A column matrix is where n = 1, as in (1.2).

A}={A11A21...Am1}. (1.2)

A row matrix is where m = 1, as in (1.3).

A⌋=⌊A11A12....A1n⌋. (1.3)

A square matrix is where m = n, as in (1.4).

A]=[A11A13....A1nA21A22....A2n.........An1An2....Ann]. (1.4)

The square matrix of (1.4) is said to be of order n.

The transpose of a matrix is obtained by interchanging its rows with its columns, i.e. the transpose of a matrix is obtained by making its first column, its first row, and its second column, its second row, and so on and so forth. For example if

A]=[0−1234−5]

then the transpose of [A] is given by

A]T=[03−142−5].

A super-matrix is a matrix whole elements themselves are matrices, as shown in (1.5).

A]=[A11A12A13......A1nA21A22A23A2nA31A32A33A3n................Am1Am2Am3......Amn].=[abcd] (1.5)

where,

a]=[A11A12A21A22][b]=[A13.......A1nA23.......A2n][c]=[A31A32........Am1Am2][d]=[A33.......A3n........Am3Amn].

The matrix of (1.5) is to be partitioned, as shown by the broken lines. Matrix partitioning is found to be a very useful aid when isolating certain physical features within the matrix.

A null matrix is one which has all its elements equal to zero.

A diagonal matrix is a square matrix where all the elements except those of the main diagonal are zero, as in (1.6).

A]=[A110.......00A2200A33...........00Ann]. (1.6)

A scalar matrix is a diagonal matrix where all the diagonal elements are equal to the same scalar quantity. When the scalar quantity is unity, the matrix is called the unit or identity matrix, and is denoted by [I].

An upper triangular matrix is a matrix which contains all its non-zero elements in and above its main diagonal, as in (1.7).

A11A12A13...A1n0A22A23...A2n00A33...A3n............000Ann]. (1.7)

A lower triangular matrix is one which contains all its non-zero elements in and below its main diagonal, as in (1.8).

A110.....0A21A22..........An1An2.....Ann]. (1.8)

A band matrix has all its non-zero elements contained in a diagonal strip as shown in (1.9) and (1.10). The centre diagonal of the strip is not necessarily the main diagonal.

A11A120000....0A21A22A23000....00A32A33A340....000A43A44A450....0.......................................000.0An,n−1An,n] (1.9)

The bandwidth of the matrix of (1.10) is said to be NW.

(1.10)

A symmetric matrix is where all

ij=Aji.

The trace of a matrix is obtained by summing all the elements on its leading diagonal, as follows:

[A]=∑i=1nAii,

and the leading diagonal of a square matrix consists of the elements A11, A22, A33,…, Ann.

1.2 ADDITION AND SUBTRACTION OF MATRICES

If

A]=[1−123−45]

and

B]=[−40−216−7]

then,

A]+[B]=[(1−4)(−1+0)(2−2)(3+1)(−4+6)(5−7)]=[−3−1042−2].

Similarly,

A]−[B]=[(1+4)(−1−0)(2+2)(3−1)(−4−6)(5+7)]=[5−142−1012].

1.3 MATRIX MULTIPLICATION

In the relationship [A][B] = [C], [A] is known as the premultiplier, [B] the postmultiplier and [C] the product. Furthermore, if [A] is of order m x n and [B] is of order n x p then [C] is of order m x p. It should be noted that [B] must always have its number of rows equal to the number of columns in [A].

If

A]=[12−103−4]

and

B]=[−230−141],

then [C] is obtained by multiplying the columns of [B] by the rows of [A], so that, in general,

ij=∑k=1nAik×Bkj,

i.e. to obtain each Cij, the ith row of [A] must be premultiplied into the jth column of [B], as follows:

C]=[1×−2+2×(−1)1×3+2×41×0+2×1−1×−2+0×(−1)−1×3+0×4−1×0+0×13×−2−4×(−1)3×3−4×41×0−4×1]=[−41122−30−2−7−4].

Similarly, if

A}={1−23}

and

B⌋=⌊−145⌋

then

A}⌊B⌋=[1×(−1)1×41×5−2×(−1)−2×4−2×53×(−1)3×43×5]=[−1452−8−10−31215]

and

B⌋{A}=(−1×1)+(4×−2)+(5×3)=−1−8+15=6.

Thus, in general, the vector product {A}[B] will...