A variant of the Cholesky decomposition is the form , where R 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:
to obtain:
a_{11} = l_{11}^{2} | ||
a_{21} = l_{21}l_{11} | ||
a_{22}=l_{21}^{2} + l^{2}_{22} | ||
a_{32}=l_{31}l_{21}+l_{32}l_{22} | l_{32}=(a_{32}-l_{31}l_{21})/l_{22}, etc. |
In general for and :
Because A is symmetric and positive definite, the expression under the square root is always positive, and all l_{ij} are real (see [Golub89]).