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:
|a11 = l112|
|a21 = l21l11|
|a22=l212 + l222|
In general for and :
Because A is symmetric and positive definite, the expression under the square root is always positive, and all lij are real (see [Golub89]).