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Cholesky decomposition
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(Definition)
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A symmetric and positive definite matrix can be efficiently decomposed into a lower and upper triangular matrix. For a matrix of any type, this is achieved by the LU decomposition which factorizes  . If  satisfies the above criteria, one can decompose more efficiently into  where  is a lower triangular matrix with positive diagonal elements.  is called the Cholesky triangle.
To solve  , one solves first  for  , and then  for  .
A variant of the Cholesky decomposition is the form  , where  is upper triangular.
Cholesky decomposition is often used to solve the normal equations in linear least squares problems; they give
 , in which  is symmetric and positive definite.
To derive  , we simply equate coefficients on both sides of the equation:
Solving for the unknowns (the nonzero  s), for
 and
 , we get:
Because  is symmetric and positive definite, the expression under the square root is always positive, and all  are real.
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- Originally from The Data Analysis Briefbook (http://rkb.home.cern.ch/rkb/titleA.html)
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"Cholesky decomposition" is owned by gufotta. [ full author list (2) | owner history (2) ]
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Cross-references: real, square root, expression, equation, sides, coefficients, equate, linear least squares, normal equations, upper triangular, diagonal, positive, lower triangular matrix, LU decomposition, type, upper triangular matrix, matrix, positive definite, symmetric
There are 4 references to this entry.
This is version 11 of Cholesky decomposition, born on 2002-01-05, modified 2006-06-04.
Object id is 1287, canonical name is CholeskyDecomposition.
Accessed 44303 times total.
Classification:
| AMS MSC: | 15-00 (Linear and multilinear algebra; matrix theory :: General reference works ) | | | 65-00 (Numerical analysis :: General reference works ) | | | 62J05 (Statistics :: Linear inference, regression :: Linear regression) |
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Pending Errata and Addenda
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