Mathematics

Square Systems

The most common situation involves a square coefficient matrix A and a single right-hand side column vector b. The solution, x = A\b, is then the same size as b. For example,

x = A\u

x =
      10
     -12
       5

It can be confirmed that A*x is exactly equal to u.

If A and B are square and the same size, then X = A\B is also that size.

X = A\B

X =
      19    -3    -1
     -17     4    13
       6     0    -6

It can be confirmed that A*X is exactly equal to B.

Both of these examples have exact, integer solutions. This is because the coefficient matrix was chosen to be pascal(3), which has a determinant equal to one. A later section considers the effects of roundoff error inherent in more realistic computation.

A square matrix A is singular if it does not have linearly independent columns. If A is singular, the solution to AX = B either does not exist, or is not unique. The backslash operator, A\B, issues a warning if A is nearly singular and raises an error condition if exact singularity is detected.

Overview

Overdetermined Systems