Singularity

Consider the system

It can be easily checked that all the following vectors are solutions to the system:

Why does this system have a non-unique solution? The third row of equals the sum of the first plus the second row, therefore is singular.

The solution of the linear system :

• nonsingular there is a unique solution ;
• singular
  • if there are infinitely many solutions,
  • if there are no solutions.

Singularity has to be detected and reported during the LU decomposition step. Numerically, if we apply the LU decomposition with pivoting to the example matrix

we end up with an upper triangular matrix which has a zero diagonal element. When back-substituting we obtain the equation , which will hold for any value . When diagonal elements of are zero (or very small) the system is (almost) singular.