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A pivot element is a fundamental concept in various algorithms, particularly in numerical linear algebra and data structures. In the context of algorithms like Gaussian elimination or quicksort, a pivot element is a chosen element in a matrix or an array that is used to simplify the remaining parts of the structure.

For example, in Gaussian elimination, a pivot element is typically a non-zero entry on the diagonal of a matrix. This element is then used to perform row operations to create zeros in the entries below it, effectively transforming the matrix into a simpler form like row echelon form. The choice of the pivot element can significantly impact the stability and efficiency of the algorithm, leading to techniques like partial pivoting where the element with the largest absolute value in a column is selected as the pivot.

In the quicksort sorting algorithm, the pivot element is an element chosen from the array. The array is then partitioned into two sub-arrays: elements smaller than the pivot and elements larger than the pivot. The pivot is placed in its final sorted position, and the process is recursively applied to the sub-arrays. The selection of the pivot strategy is crucial for quicksort's average-case performance.