Important: The linalg package has been deprecated. Use the superseding command LinearAlgebra[LUDecomposition], instead.

- For information on migrating linalg code to the new packages, see examples/LinearAlgebraMigration.

The routine cholesky computes the cholesky decomposition of the matrix A.

The result is a lower triangular matrix R such that $R\mathrm{transpose}\left(R\right)=A$.

This decomposition assumes the matrix A is positive-definite. I.e. R exists with real elements on the diagonal.  cholesky will fail with an error when called on a demonstrably non-positive-definite matrix.

$\det \left(A\right)={\left(\prod _{i=1}^n{R}_{i,i}\right)}^2$.

The command with(linalg,cholesky) allows the use of the abbreviated form of this command.

$\mathrm{with}\left(\mathrm{linalg}\right)\:$

$S≔\mathrm{matrix}\left(3\,3\,\left[1\,2\,3\,0\,1\,1\,0\,0\,4\right]\right)$

$S≔\left[\begin{array}{ccc}1& 2& 3\\ 0& 1& 1\\ 0& 0& 4\end{array}\right]$

$A≔\mathrm{evalm}\left(\mathrm{transpose}\left(S\right)\&*S\right)$

$A≔\left[\begin{array}{ccc}1& 2& 3\\ 2& 5& 7\\ 3& 7& 26\end{array}\right]$

$R≔\mathrm{cholesky}\left(A\right)$

$R≔\left[\begin{array}{ccc}1& 0& 0\\ 2& 1& 0\\ 3& 1& 4\end{array}\right]$