A pivot element, often shortened to pivot, is a nonzero entry of a matrix selected during Gaussian elimination as the leading entry of a row and used to eliminate other entries in its column. Its location is called a pivot position. A column containing a pivot is a pivot column, and the corresponding variable in a linear system of equations is a pivot variable (Shoup 2009).
The number of pivot elements in an echelon form is the matrix rank. When the pivot-column indices are applied to the original matrix, the corresponding columns form a basis for its column space (Piziak and Odell 1999). Columns without pivots correspond to free variables. In numerical computation, pivoting uses row or column interchanges to select a suitable pivot (Golub and Van Loan 2013).
