An indeterminate system is a system of simultaneous equations (especially linear equations) which has more than one solution. The system may be said to be underspecified. If the system is linear, then the presence of more than one solution implies that there is an infinite number of solutions, but that property does not extend to nonlinear systems.
An indeterminate system is consistent, the latter implying that there exists at least one solution. For a system of linear equations, the number of equations in an indeterminate system could be the same as the number of unknowns, less than the number of unknowns (an underdetermined system), or greater than the number of unknowns (an overdetermined system). Conversely, any of those three cases may or may not be indeterminate.
The following examples of indeterminate systems of equations have respectively fewer equations than, as many equations as, and more equations than unknowns:
Conditions giving rise to indeterminacy
In linear systems, indeterminacy occurs if and only if the number of independent equations (the rank of the augmented matrix of the system) is less than the number of unknowns and is the same as the rank of the coefficient matrix. For if there are at least as many independent equations as unknowns that will eliminate any stretches of overlap of the equations' surfaces in the geometric space of the unknowns (aside from possibly a single point), so there cannot be more than one solution; and if the rank of the augmented matrix exceeds (necessarily by one if at all) the rank of the coefficient matrix then the equations jointly contradict each other, so there are no solutions.
Finding the solution set of an indeterminate linear system
Let the system of equations be written in matrix form as
where A is the m×n coefficient matrix, x is the n×1 vector of unknowns, and b is an m×1 vector of constants. Then if the system is indeterminate, the infinite solution set is the set of all x vectors generated by
where is the Moore-Penrose pseudoinverse of A and w is any n×1 vector.
- James, M., "The generalised inverse", Mathematical Gazette 62, June 1978, 109–114.