Mathematics    

Square Systems

The most common situation involves a square coefficient matrix A and a single right-hand side column vector b.

Nonsingular Coefficient Matrix

If the matrix A is nonsingular, the solution, x = A\b, is then the same size as b. For example,

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.

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 computations.

Singular Coefficient Matrix

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 it detects exact singularity.

If A is singular and AX = b has a solution, you can find a particular solution that is not unique, by typing

P is a pseudoinverse of A. If AX = b does not have an exact solution, pinv(A) returns a least-squares solution.

For example,

is singular, as you can verify by typing

Exact Solutions.   For b =[5;2;12], the equation AX = b has an exact solution, given by

You can verify that pinv(A)*b is an exact solution by typing

Least Squares Solutions.   On the other hand, if b = [3;6;0], then AX = b does not have an exact solution. In this case, pinv(A)*b returns a least squares solution. If you type

you do not get back the original vector b.

You can determine whether AX = b has an exact solution by finding the row reduced echelon form of the augmented matrix [A b]. To do so for this example, type

Since the bottom row contains all zeros except for the last entry, the equation does not have a solution. In this case, pinv(A) returns a least-squares solution.


  General Solution Overdetermined Systems