6.5.3.1. Numerical Examples

6. Process or Product Monitoring and Control
6.5. Tutorials
6.5.3. Elements of Matrix Algebra

6.5.3.1. Numerical Examples

Numerical examples of matrix operations

Numerical examples of the matrix operations described on the previous page are given here to clarify these operations.

Sample matrices

then

Matrix addition, subtraction, and multipication

A + B = [8 8; 4 12] and
A - B = [2 4; 2 2]

and

A*B = [21 40; 16 41] and B*A = [21 32; 20 41]

Multiply matrix by a scalar

To multiply a a matrix by a given scalar, each element of the matrix is multiplied by that scalar 2*A = [10 12; 6 14] and 0.5*B = [1.5 1.0; 0.5 2.5]

Pre-multiplying matrix by transpose of a vector

Pre-multiplying a p x n matrix by the transpose of a p-element vector yields a n-element transpose c' = a'*B = [a1 a2]*[b11 b12 b13; b21 b23 b33]' = [c1 c2 c3]

Post-multiplying matrix by vector

Post-multiplying a p x n matrix by an n-element vector yields an n-element vector c = B*A = [b11 b12 b13; b21 b22 b23]*[a1 a2 a3] = [c1 c2]

Quadratic form

It is not possible to pre-multiply a matrix by a column vector, nor to post-multiply a matrix by a row vector. The matrix product a'Ba yields a scalar and is called a quadratic form. Note that B must be a square matrix if a'Ba is to conform to multiplication. Here is an example of a quadratic form a'*B*a = [2 3]*[1 2; 3 1]*[2 3] = [11 7]*[2 3] = 43

a'*B*a = [2 3]*[1 2; 3 1]*[2 3] = [11 7]*[2 3] = 43

Inverting a matrix

The matrix analog of division involves an operation called inverting a matrix. Only square matrices can be inverted. Inversion is a tedious numerical procedure and it is best performed by computers. There are many ways to invert a matrix, but ultimately whichever method is selected by a program is immaterial. If you wish to try one method by hand, a very popular numerical method is the Gauss-Jordan method.

Identity matrix

To augment the notion of the inverse of a matrix, A^-1 (A inverse) we notice the following relation

A^-1A = A A^-¹= I

I is a matrix of form I =
[1 0 0 ... 0;
0 1 0 ... 0;
0 0 1 ... 0;
.... ;
0 0 0 1]

I is called the identity matrix and is a special case of a diagonal matrix. Any matrix that has zeros in all of the off-diagonal positions is a diagonal matrix.