Pivot

Definition

Let $F$ be a field and let $A$ be a matrix over $F$ in row echelon form. A pivot of $A$ is the leading entry of a nonzero row of $A$, that is, the first nonzero entry of that row. A pivot column of $A$ is a column that contains a pivot. For a general matrix $B$, take a row echelon form of $B$ that is row equivalent to $B$. The pivot positions of $B$ are the positions of the pivots of this form.

Remarks

The pivot positions do not depend on the choice of the row echelon form. The number of pivots is the rank of the matrix. In the system $Ax = b$, the unknowns of the pivot columns are the basic variables, and the other unknowns are the free variables. In numerical work, partial pivoting chooses as pivot an entry of largest absolute value in its column, to reduce rounding errors.

Explanations

L1 human

A pivot is the first nonzero number in a row of a staircase matrix. During elimination we divide by the pivot and use it to clear the entries below it. Each pivot fixes one unknown, and each column without a pivot gives a free unknown.