Pivot element ( $a_{kk}^{(k - 1)}$ )

A pivot element is the current diagonal entry used to eliminate entries below it in Gaussian elimination, after any row interchange. At step

k

it is

a_{kk}^{(k - 1)}

, where

A^{(k - 1)}

is the current reduced matrix. Computing elimination multipliers requires this entry to be nonzero. A zero pivot element can reflect an unsuitable row order even for an invertible matrix, as Pivoted LU decomposition illustrates.

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A pivot element refers to a particular value or position within a data structure that serves a crucial role during various algorithms, notably in sorting and optimization contexts. The specific meaning of "pivot" can vary depending on the context in which it is used. Here are a few common scenarios: 1. **In QuickSort Algorithm**: The pivot element is the value used to partition the array into two sub-arrays.