Cholesky factorization
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:'
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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.
References
| [a1] | N.J. Higham, "Accuracy and stability of numerical algorithms" , SIAM (Soc. Industrial Applied Math.) (1996) |
Cholesky factorization. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Cholesky_factorization&oldid=37467