7.7   MATRICES, NOTATION, AND COMPUTING

In a computer program, rather than using separate variables x, y, u, v, and so
on, it is more convenient mathematically to use “subscripted variables” as the
elements of a vector.

The entire vector is then represented by the single symbol x, which is made
up of several elements x₁, x₂, and so on.

Matrices are now made up of elements with two suffices:

In a computer program, the subscripts appear in brackets, so that a vector
could be represented by the elements X(1), X(2), and X(3), while the elements
of the matrix are A(1,1), A(1,2), and so on.

It is in matrix operations that this notation really earns its keep. Suppose
that we have a relationship

where the vectors have three elements and the matrix is 3 × 3. Instead of a
massive block of arithmetic, the entire product is expressed in just five lines
of Basic program:

FOR I=1 TO 3
  X(I)=0
  FOR J=1 TO 3
    X(I)=X(I)+T(I, J)*U(J)
  NEXT J
NEXT I

For the matrix product C = AB, the program is hardly any more complex:

FOR I=1 TO 3
  FOR J=1 TO 3
   C(I,J)=0
   FOR K=1 TO 3
    C(I,J)=C(I,J)+A(I,K)*B(K,J)
   NEXT K
  NEXT J
NEXT I

Or in Java or C it becomes

for(i = 1; i<=3; i++){
  for(j = 1; j<=3; j++) {
    c[i][j] = 0;
    for(k = 1; k<=3;k++) {
      c[i][j] += a[i][k]*b[k][j];
    }
  }
}

These examples would look almost identical in a variety of languages and
would show the same economy of programming effort.

In Matlab the shorthand of matrix operations goes even farther—but there
is a danger that the engineroom will be lost to view behind the paintwork.

But beware!!