Indeterminate system

In mathematics, particularly in number theory, an indeterminate system has fewer equations than unknowns but an additional a set of constraints on the unknowns, such as restrictions that the values be integers.

Examples

Linear indeterminate equations

For given integers a, b and n, a linear indeterminant equation is $$ax+by=n$$ with unknowns x and y restricted to integers. The necessary and sufficient condition for solutions is that the greatest common divisor, $(a,b)$, is divisable by n.

Finding the solution set of an indeterminate linear system

Let the system of equations be written in matrix form as

$Ax=b$

where $A$ is the $m\times n$ coefficient matrix, $x$ is the $n\times 1$ vector of unknowns, and $b$ is an $m\times 1$ vector of constants. In which case, if the system is indeterminate, then the infinite solution set is the set of all $x$ vectors generated by

$x=A^{+}b+[I_{n}-A^{+}A]w$

where $A^{+}$ is the Moore–Penrose pseudoinverse of $A$ and $w$ is any $n\times 1$ vector.

See also

• Indeterminate equation
• Indeterminate form
• Indeterminate (variable)
• Linear algebra
• Simultaneous equations
• Independent equation
• Identifiability

Further reading

• Lay, David (2003). Linear Algebra and Its Applications. Addison-Wesley. ISBN 0-201-70970-8.