Row echelon form

Row echelon form

In linear algebra a matrix is in row echelon form if
* All nonzero rows are above any rows of all zeroes, and
* The leading coefficient of a row is always strictly to the right of the leading coefficient of the row above it.

This is the definition used in this article, but some texts add a third condition:
* The leading coefficient of each nonzero row is one. [See, for instance, Larson and Hostetler, "Precalculus", 7th edition.]

A matrix is in reduced row echelon form (also called row canonical form) if it satisfies the above three conditions, and if, in addition
* Every leading coefficient is the only nonzero entry in its column.

The first non-zero entry in each row is called a pivot.

Examples

This matrix is in reduced row echelon form:

:egin{bmatrix}0 & 1 & 4 & 0 & 0 \0 & 0 & 0 & 1 & 0 \0 & 0 & 0 & 0 & 1 \0 & 0 & 0 & 0 & 0 \end{bmatrix}

The following matrix is also in row echelon form, but not in reduced row form:

:egin{bmatrix}1 & 1 & 1 & 1 \0 & 9 & 0 & 2 \0 & 0 & 0 & 3 \end{bmatrix}

However, this matrix is not in row echelon form, as the leading coefficient of row 3 is not strictly to the right of the leading coefficient of row 2.

:egin{bmatrix}1 & 2 & 3 & 4 \0 & 3 & 7 & 2 \0 & 2 & 0 & 0 \end{bmatrix}

Non-uniqueness

Every non-zero matrix can be reduced to an infinite number of echelon forms (they can all be multiples of each other, for example) via elementary matrix transformations. However, all matrices and their row echelon forms correspond to exactly one matrix in reduced row echelon form.

Systems of linear equations

A system of linear equations is said to be in "row echelon form" if its augmented matrix is in "row echelon form". Similarly, a system of equations is said to be in "reduced row echelon form" or "canonical form" if its "augmented matrix" is in "reduced row echelon form".

Pseudocode

The following pseudocode converts a matrix to reduced row-echelon form:

ToReducedRowEchelonForm(Matrix M) Let "lead" = 0 Let "rowCount" be the number of rows in M Let "columnCount" be the number of columns in M FOR "r" = 0 to "rowCount" - 1 IF "columnCount" <= "lead" STOP END IF Let "i" = "r" WHILE M ["i", "lead"] = 0 Increment "i" IF "rowCount" = "i" Let "i" = "r" Increment "lead" IF "columnCount" = "lead" STOP END IF END IF END WHILE Swap rows "i" and "r" Divide row "r" by M ["r", "lead"] FOR all rows "i", from 0 to number of rows, every row except "r" Subtract M [i, lead] multiplied by row "r" from row "i" END FOR Increment "lead" END FOR END ToReducedRowEchelonForm

ee also

*Gaussian elimination
*Gauss–Jordan elimination

Notes


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