Cholesky decomposition
Topic of Estimation and inference for the multivariate normal, part of Statistical learning
Cholesky decomposition
| Type | Topic |
| Taught in | Estimation and inference for the multivariate normal |
| Part | Statistical learning |
| Questions using it | 1 |
The factorisation \Sigma=LL^\top of a positive-definite matrix with L lower triangular. If \mathbf{Z}\sim\mathrm{N}_p(\mathbf{0},I) then \boldsymbol\mu+L\mathbf{Z}\sim\mathrm{N}_p(\boldsymbol\mu,\Sigma), which is how correlated normal vectors are drawn.
In Estimation and inference for the multivariate normal
so X is standard normal. This is the inverse-transform method. Draw p independent copies and stack them as \mathbf{Z}\sim\operatorname{N}_p(\mathbf{0},I_p). Any matrix B with BB^\top=\Sigma then has the covariance required by the linear-image formula. The spectral factor B=\Gamma\Lambda^{1/2}, whose columns are eigenvectors scaled by square-root eigenvalues, is one such factor, and the Cholesky factor is another. Set
\mathbf{X}=\boldsymbol{\mu}+B\mathbf{Z}.
Used in
- Taught in Estimation and inference for the multivariate normal
- Questions that use it (1) — Two factors of one covariance
- All 9 questions on Estimation and inference for the multivariate normal
See also
Positive definite matrix · Invariance of the maximum likelihood estimator · Wishart distribution · Sampling law of the normal mean and scatter · Multivariate central limit theorem · The limit of Hotelling’s statistic · Null distribution of Hotelling’s · Two-sample Hotelling test