row reduction

Row reduction, also known as Gaussian elimination, is an algorithm for solving a system of linear equations

To describe row reduction, it is convenient to formulate a linearsystem as a single matrix-vector equation $Ax=b,$ where

are, respectively, the $n\times m$ matrix of coefficients of thelinear system, the $n$-place column vector of the scalars from theright-hand of the equations, and the $m$-place column vector ofunknowns.

The method consists of combining the coefficient matrix $A$ with theright hand vector $b$ to form the “augmented” $n\times (m+1)$ matrix

where each $R_i$ is the $m+1$-place row vector corresponding to row$i$ of the augmented matrix.

A sequence of elementary row operations is then applied to this matrixso as to transform it to row echelon form. The elementary operations are:

• row scaling: the multiplication a row by a nonzeroscalar;

  $$R_i\mapsto c{R}_i,c\ne 0;$$

• row exchange: the exchanges of two rows;

  $$R_i\leftrightarrow R_j;$$

• row replacement: the addition of a multiple of one row toanother row;

  $$R_i\mapsto R_i+c{R}_j,c\ne 0,i\ne j.$$

Note that these operations are “legal” because $x$ is a solution ofthe transformed system if and only if it is a solution of the initial system.

If the number of equations equals the number of variables ($m=n$), andif the coefficient matrix $A$ is non-singular (http://planetmath.org/singular), then the algorithm willterminate when the augmented matrix has the following form:

With these assumptions, there exists a unique solution, which can beobtained from the above matrix by back substitution.

For the general case, the termination procedure is somewhat morecomplicated. First recall that a matrix is in echelon form if eachrow has more leading zeros than the rows above it. A pivot is theleading non-zero entry of some row. We then have

• If there is a pivot in the last column, the system isinconsistent ; there will be no solutions.

• If that is not the case, then the general solution will have $d$degrees of freedom, where $d$ is the number of columns from $1$ to$m$ that have no pivot. To be more precise, the general solutionwill have the form of one particular solution plus an arbitrary linearcombination of $d$ linearly independent $n$-vectors.

  In even more prosaic language, the variables in the non-pivotcolumns are to be considered “free variables” and should be“moved” to the right-hand side of the equation. The generalsolution is then obtained by arbitrarily choosing values of the freevariables, and then solving for the remaining “non-free” variablesthat reside in the pivot columns.

A variant of Gaussian elimination is Gauss-Jordan elimination. Inthis variation we reduce to echelon form, and then if the systemproves to be consistent, continue to apply the elementary rowoperations until the augmented matrix is in reduced echelonform. This means that not only does each pivot have all zeroesbelow it, but that each pivot also has all zeroes above it.

In essence, Gauss-Jordan elimination performs the back substitution;the values of the unknowns can be read off directly from the terminalaugmented matrix. Not surprisingly, Gauss-Jordan elimination isslower than Gaussian elimination. It is useful, however, for solvingsystems on paper.

Title	row reduction
Canonical name	RowReduction
Date of creation	2013-03-22 12:06:48
Last modified on	2013-03-22 12:06:48
Owner	rmilson (146)
Last modified by	rmilson (146)
Numerical id	18
Author	rmilson (146)
Entry type	Algorithm
Classification	msc 15A06
Synonym	Gaussian elimination
Synonym	Gauss-Jordan elimination
Related topic	RowEchelonForm
Related topic	ReducedRowEchelonForm
Related topic	ElementaryMatrix
Defines	pivot
Defines	row operation
Defines	row exchange
Defines	row replacement
Defines	row scaling