Encyclopedia > Matrix inversion

  Article Content

Matrix inversion

Matrix inversion is the following problem in linear algebra: given a square n-by-n matrix A, find a square n-by-n matrix B (if one exists) such that AB = In, the n-by-n identity matrix.

The Gauss-Jordan elimination is an algorithm that can be used to determine whether a given matrix is invertible and to find the inverse. An alternative is the Cholesky decomposition which generates two upper triangular matrices which are easier to invert. For special purposes, it may be convenient to invert matrices by treating mn-by-mn matrices as m-by-m matrices of n-by-n matrices, and applying one or another formula recursively (other sized matrices can be padded out with dummy rows and columns). For other purposes, a variant of Newton's method may be convenient (particularly when dealing with families of related matrices, so inverses of earlier matrices can be used to seed generating inverses of later matrices).

Writing another special matrix of cofactors, known as an adjoint matrix, can also be an efficient way to calculate the inverse of small matrices (since this method is essentially recursive, it becomes inefficient for large matrices). To determine the inverse, we calculate a matrix of cofactors:

<math>A^{-1}={1 \over \begin{vmatrix}A\end{vmatrix}}\left(C_{ij}\right)^{T}={1 \over \begin{vmatrix}A\end{vmatrix}}
\begin{pmatrix} C_{11} & C_{21} & \cdots & C_{j1} \\ C_{12} & \ddots & \vdots & C_{j2} \\ \vdots & \cdots & \ddots & \vdots \\ C_{1i} & \cdots & \cdots & C_{ji} \\ \end{pmatrix}</math> where <math>\begin{vmatrix}A\end{vmatrix}</math> is the determinant of A, <math>C_{ij}</math> is the matrix cofactor, and <math>A^{T}</math> represents the matrix transpose.

In most practical applications, it is in fact not necessary to invert a matrix, but only to solve a system of linear equations. Various fast algorithms for special classes of such systems have been developed.



All Wikipedia text is available under the terms of the GNU Free Documentation License

 
  Search Encyclopedia

Search over one million articles, find something about almost anything!
 
 
  
  Featured Article
Kings Park, New York

... and the average family size is 3.32. In the town the population is spread out with 25.2% under the age of 18, 5.7% from 18 to 24, 31.9% from 25 to 44, 23.3% from 45 to 64, ...

 
 
 
This page was created in 42.3 ms