Square Systems :: Matrices and Linear Algebra (Mathematics)

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,

A = pascal(3);
u = [3; 1; 4];
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.

B = magic(3);
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 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 = pinv(A)*b
```

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

For example,

A = [ 1     3     7
     -1     4     4
      1    10    18 ]

is singular, as you can verify by typing

```
det(A)

ans =
     0
```

Note For information about using pinv to solve systems with rectangular coefficient matrices, see Pseudoinverses.

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

pinv(A)*b

ans =
    0.3850
   -0.1103
    0.7066

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

A*pinv(A)*b

ans =
    5.0000
    2.0000
   12.0000

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

A*pinv(A)*b

ans =
   -1.0000
    4.0000
    2.0000

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, enter

rref([A b])
ans =
    1.0000         0    2.2857         0
         0    1.0000    1.5714         0
         0         0         0    1.0000

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