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, type

• 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