** Next:** Multiple systems with the
** Up:** Linear Systems of Algebraic
** Previous:** Operation count
** Contents**

`
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.
`

** Next:** Multiple systems with the
** Up:** Linear Systems of Algebraic
** Previous:** Operation count
** Contents**
Adrian Sandu
2001-08-26