nulpivots

Nulpivots are a concept used in linear algebra and numerical analysis to describe pivot positions that cannot serve as leading variables during matrix factorization because the candidate pivot is zero or numerically indistinguishable from zero. In the process of Gaussian elimination, a pivot is a nonzero entry chosen from a column to eliminate entries below it. A nulpivot occurs when, at a particular step, the value available to act as a pivot is zero (or below a specified tolerance in floating point arithmetic), signaling a dependent row or a zero row and indicating that the corresponding column does not contribute a basic variable.

Detection and interpretation: In exact arithmetic, a true zero pivot reveals that the matrix has deficient rank and that the associated column does not provide a new leading variable. In numerical practice, small pivots are treated as zeros within a tolerance, which makes nulpivots a practical signal of degeneracy or near-dependence. Rank-revealing techniques such as QR factorization with pivoting, complete pivoting, or singular value decomposition are commonly used to identify null directions and to separate basic from free variables.

Mathematical significance: Nulpivots mark the boundary between the row space and the null space of a matrix. They indicate that some variables are free and that the solution set to a linear system has dimensions equal to the nullity of the matrix. They also influence the conditioning of problems and the design of stable numerical solvers.

Examples and applications: A matrix with a row of zeros after elimination exhibits nulpivots in the corresponding columns, reflecting a nontrivial null space. Understanding nulpivots is helpful in solving ill-conditioned systems, performing least-squares computations, and analyzing control systems where redundancy must be managed.

See also: null space, pivot, rank, QR decomposition, SVD.