- Gauss–Jordan elimination
In
linear algebra , Gauss–Jordan elimination is a version ofGaussian elimination that puts zeros both above and below eachpivot element as it goes from the top row of the given matrix to the bottom. In other words, Gauss-Jordan elimination brings a matrix toreduced row echelon form , whereas Gaussian elimination takes it only as far asrow echelon form . Every matrix has a reduced row echelon form, and this algorithm is guaranteed to produce it.Gauss–Jordan elimination is considerably less efficient than Gaussian elimination with
backsubstitution when solving asystem of linear equations . However, it is well suited for calculating thematrix inverse .It is named in honor of
Carl Friedrich Gauss andWilhelm Jordan .Application to finding inverses
If Gauss–Jordan elimination is applied on a
square matrix , it can be used to calculate the matrix's inverse. This can be done by augmenting the square matrix with theidentity matrix of the same dimensions, and through the following matrix operations::If the original square matrix, , is given by the following expression::
Then, after augmenting by the identity, the following is obtained::
By performing
elementary row operations on the matrix until reachesreduced row echelon form , the following is the final result::
The matrix augmentation can now be undone, which gives the following::
A matrix is non-singular (meaning that it has an inverse matrix)
iff the identity matrix can be obtained using only elementary row operations.References
* Lipschutz, Seymour, and Lipson, Mark. "Schaum's Outlines: Linear Algebra". Tata McGraw-hill edition. Delhi 2001. pp. 69-80.
*External links
* [http://users.powernet.co.uk/kienzle/octave/matcompat/scripts/linear-algebra/rref.m Algorithm for Gauss-Jordan elimination in Matlab]
* [http://elonen.iki.fi/code/misc-notes/python-gaussj/index.html Algorithm for Gauss-Jordan elimination in Python]
* [http://lipe.advant.com.br/unicenp/gauss-jordan.php An online tool solve nxm linear systems using Gauss-Jordan elimination (source-code and mobile version included), by Felipe Santos de Andrade]
* [http://www.cs.berkeley.edu/~wkahan/MathH110/gji.pdf Algorithm for Gauss-Jordan elimination in Basic]
* [http://math.fullerton.edu/mathews/n2003/GaussianJordanMod.html Module for Gauss-Jordan Elimination]
* [http://vivaldi.ucsd.edu:8080/~kcheng/ece155/hwsoln/Gaussian-Jordan.pdf Example of Gauss-Jordan Elimination "Step-by-Step"]
Wikimedia Foundation. 2010.