Page 82 - Standard Handbook Of Petroleum & Natural Gas Engineering

Numerical Methods 71

Solution of Sets of Simultaneous Linear Equations

A matrix is a rectangular array of numbers, its size being determined by the
number of rows and columns in the array. In this context, the primary concern
is with square matrices, and matrices of column dimension 1 (column vectors)
and row dimension 1 (row vectors).
Certain configurations of square matrices are of particular interest. If

the diagonal consisting of cl,, cz2, c33 and cqq is the main diagonal. The matrix is
symmetric if c = c,,. If all elements below the main diagonal are zero (blank), it
is an upper t&ngular matrix, while if all elements above the main diagonal are
zero, it is a lower triangular matrix. If all elements are zero except those on the
main diagonal, the matrix is a diagonal matrix and if a diagonal matrix has all
ones on the diagonal, it is the unit, or identity, matrix.
Matrix addition (or subtraction) is denoted as S = A + B and defined as

s. = a,. + b.
II ?I 'J
where A. B, and S have identical row and column dimensions. Also,

A+B=B+A
A-B=-B+A

Matrix multiplication, represented as P = AB, is defined as

where n is the column dimension of A and the row dimension of B. P will have
row dimension of A and column dimension of B. Also
AI = A

and
IA = A

while, in general,
AB # BA

Matrix division is not defined, although if C is a square matrix, C-' (the inverse
of C) can usually be defined so that
CC-' = I