Matrix Operations

  We have already seen the matrix operator ' (prime) for transposing matrices. The arithmetic operators presented in Section 5 also operate on matrices. In each case the operator behaves in a manner consistent with standard linear algebra practises.

The operators + and - permit the addition and subtraction of matrices, and are defined whenever the matrices have the same dimension. For example, the following is a valid expression.
>> [1 2 3; 4 5 6] + [3 2 1; 1 1 1];
The exception to this rule is the addition (subtraction) of a scalar to (from) a matrix. In this case the scalar is added to or subtracted from each element of the matrix individually.
>> [1 2 3] + 1

ans =
     2     3     4

The multiplication of two matrices, denoted by A*B, is defined whenever the inner dimensions of the operands A and B are equal. For example, if
C=[1 2 3; 4 5 6], D=[1 1 1; 2 2 2], x=[1 1 1]'
then C*x, x'*x (an inner product), x*x' (an outer product) and C*D' are defined, but C*D is not. (Give these examples a try and be sure you understand how MATLAB interprets them.) In the special case when one of the operands is a scalar, each element of the matrix is multiplied by the scalar.

It is now time to explain the reason for MATLAB's two division operators. If A is a square nonsingular matrix then A\B and B/A formally correspond to the left and right multiplication of B by A . (Note: MATLAB does not actually compute the inverse of A when evaluating these expressions.) These expressions are used to solve the following types of systems of equations.

left division:

x = A\B solves A*X = B

right division:

x = A/B solves X*A = B

In CPSC 303 we will typically use left division. Whenever B has as many rows as A, left division is defined. If A is a square nonsingular matrix and b is a vector with as many rows, MATLAB evaluates the expression x = A\b (the solution to Ax = b) by factoring A with Gaussian elimination and then solving two triangular systems to compute x. When A is not square, MATLAB factors A using Householder orthogonalization and the factors are used to solve the under-determined or over-determined system of equations in the least squares sense. This can lead to surprising results if the wrong slash is used or if the dimensions of your matrices are wrong.

Finally, the expression A^p raises A to the p power. This operation is only defined if A is square and p is an scalar. For example, A^2 is equivalent to A*A, although MATLAB does not always compute powers with simple matrix multiplication.