A symmetric matrix (cf. also Symmetric matrix) is positive definite if the quadratic form is positive for all non-zero vectors or, equivalently, if all the eigenvalues of are positive. Positive-definite matrices have many important properties, not least that they can be expressed in the form for a non-singular matrix . The Cholesky factorization is a particular form of this factorization in which is upper triangular with positive diagonal elements, and it is usually written as . In the case of a scalar (), Cholesky factorization corresponds to the fact that a positive number has a positive square root. The factorization is named after A.-L. Cholesky, a French military officer involved in geodesy. It is closely connected with the solution of least-squares problems (cf. also Least squares, method of), since the normal equations that characterize the least-squares solution have a symmetric positive-definite coefficient matrix.
The Cholesky factorization can be computed by a form of Gaussian elimination that takes advantage of the symmetry and definiteness. Equating elements in the equation gives
These equations can be solved to yield a column at a time, according to the following algorithm:
It is the positive definiteness of that guarantees that the argument of the square root in this algorithm is always positive.
It can be shown [a1] that in floating-point arithmetic the computed solution satisfies , where , with a constant of order and the unit round-off (or machine precision). Here, and . This excellent numerical stability is essentially due to the equality , which guarantees that is of bounded norm relative to .
A variant of Cholesky factorization is the factorization where is unit lower triangular and is diagonal. This factorization exists for definite matrices and some (but not all) indefinite ones. When is positive definite, the Cholesky factor is given by .
A symmetric matrix is positive semi-definite if the quadratic form is non-negative for all ; thus, may be singular (cf. also Matrix). For such matrices a Cholesky factorization exists, but may not display the rank of . However, by introducing row and column permutations it is always possible to obtain a factorization
where is a permutation matrix, is upper triangular with positive diagonal elements, and .
Cf. Matrix factorization for more details and references.
|[a1]||N.J. Higham, "Accuracy and stability of numerical algorithms" , SIAM (Soc. Industrial Applied Math.) (1996)|
Cholesky factorization. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Cholesky_factorization&oldid=37467