Editing Cholesky decomposition (section)

===Monte Carlo simulation===
The Cholesky decomposition is commonly used in the [[Monte Carlo method]] for simulating systems with multiple correlated variables. The [[covariance matrix]] is decomposed to give the lower-triangular {{math|'''L'''}}. Applying this to a vector of uncorrelated observations in a sample {{math|'''u'''}} produces a sample vector '''Lu''' with the covariance properties of the system being modeled.<ref name="Matlab documentation">[http://www.mathworks.com/help/techdoc/ref/randn.html Matlab randn documentation]. mathworks.com.</ref>

The following simplified example shows the economy one gets from the Cholesky decomposition:  suppose the goal is to generate two correlated normal variables <math display=inline>x_1</math> and <math display=inline>x_2</math> with given correlation coefficient <math display=inline>\rho</math>. To accomplish that, it is necessary to first generate two uncorrelated Gaussian random variables <math display=inline>z_1</math> and <math display=inline>z_2</math> (for example, via a [[Box–Muller transform]]).  Given the  required correlation coefficient <math display=inline>\rho</math>, the correlated normal variables can be obtained via the transformations <math display=inline>x_1 = z_1</math> and <math display="inline">x_2 =  \rho z_1 + \sqrt{1 - \rho^2} z_2</math>.