linalg(deprecated)/cholesky - Maple Help

linalg(deprecated)

 cholesky
 Cholesky decomposition of a matrix

 Calling Sequence cholesky(A)

Parameters

 A - square, positive definite matrix

Description

 • 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.
 • $\mathrm{det}\left(A\right)={\left(\prod _{i=1}^{n}{R}_{i,i}\right)}^{2}$.
 • The command with(linalg,cholesky) allows the use of the abbreviated form of this command.

Examples

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

 > $\mathrm{with}\left(\mathrm{linalg}\right):$
 > $S≔\mathrm{matrix}\left(3,3,\left[1,2,3,0,1,1,0,0,4\right]\right)$
 ${S}{≔}\left[\begin{array}{ccc}{1}& {2}& {3}\\ {0}& {1}& {1}\\ {0}& {0}& {4}\end{array}\right]$ (1)
 > $A≔\mathrm{evalm}\left(\mathrm{transpose}\left(S\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&*\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}S\right)$
 ${A}{≔}\left[\begin{array}{ccc}{1}& {2}& {3}\\ {2}& {5}& {7}\\ {3}& {7}& {26}\end{array}\right]$ (2)
 > $R≔\mathrm{cholesky}\left(A\right)$
 ${R}{≔}\left[\begin{array}{ccc}{1}& {0}& {0}\\ {2}& {1}& {0}\\ {3}& {1}& {4}\end{array}\right]$ (3)