On the existence and computation of $LU$-factorizations with small pivots

Let A be an n by n matrix which may be singular with a one-dimensional null space, and consider the LU-factorization of A. When A is exactly singular, we show conditions under which a pivoting strategy will produce a zero nth pivot. When A is not singular, we show conditions under which a pivoting strategy will produce an nth pivot that is $O({\sigma _n})$ or $O({\kappa ^{ - 1}}(A))$, where ${\sigma _n}$ is the smallest singular value of A and $\kappa (A)$ is the condition number of A. These conditions are expressed in terms of the elements of ${A^{ - 1}}$ in general but reduce to conditions on the elements of the singular vectors corresponding to ${\sigma _n}$ when A is nearly or exactly singular. They can be used to build a 2-pass factorization algorithm which is guaranteed to produce a small nth pivot for nearly singular matrices. As an example, we exhibit an LU-factorization of the n by n upper triangular matrix \[ T = \left [ {\begin {array}{*{20}{c}} 1 & { - 1} & {} & { - 1} & {} & \cdot & \cdot & { - 1} \\ {} & 1 & {} & \cdot & {} & {} & {} & \cdot \\ {} & {} & {} & 1 & \cdot & {} & {} & {} \\ {} & {} & {} & {} & \cdot & \cdot & {} & \cdot \\ {} & {} & 0 & {} & {} & \cdot & \cdot & \cdot \\ {} & {} & {} & {} & {} & {} & \cdot & { - 1} \\ {} & {} & {} & {} & {} & {} & {} & 1 \\ \end {array} } \right ]\] that has an nth pivot equal to $2^{-(n-2)}$.