Linear and multilinear algebra; matrix theory Definition

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.

Reading path

To understand this page from the ground up, read these 3 pages in order. The foundations come first.

Show the path as a graph

Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.

no evidence level

Matrix definition
Row echelon form definition
Pivot definition

Tag	`0BCMGDHZGR`
URI	`m2://0BCMGDHZGR`
Kind	definition
Writing	W1 stated
MSC	15A06
Defines	Pivot
Uses	Row echelon form

Pivot

Definition

Remarks

Reading path

Further reading

What this page needs for W2

What links here