Positive definite matrix
Topic of Covariance and the multivariate normal, part of Statistical learning
Positive definite matrix
| Type | Topic |
| Taught in | Covariance and the multivariate normal |
| Part | Statistical learning |
| Questions using it | 10 |
A symmetric matrix \Sigma with \mathbf{a}^\top\Sigma\mathbf{a}>0 for every non-zero \mathbf{a}; positive semi-definite when \ge0. Every covariance matrix is positive semi-definite, because \mathbf{a}^\top\Sigma\mathbf{a} is the variance of \mathbf{a}^\top\mathbf{X}.
In Covariance and the multivariate normal
Symmetry of a proposed covariance is necessary and is not enough: every linear combination must have a nonnegative variance. Sylvester’s criterion turns the quadratic-form condition into determinants. Strict positive definiteness requires every leading principal minor to be positive. Positive semi-definiteness requires every principal minor, not only the leading ones, to be nonnegative. A matrix can pass a leading-minor check aimed at the wrong one of these two conclusions and still fail to be a covariance.
Used in
- Taught in Covariance and the multivariate normal
- Also used in Joint distributions · Functions of random variables and limit theorems · Bayesian linear regression · Estimation and inference for the multivariate normal · Principal component analysis · Mixtures and the EM algorithm · Model selection for stochastic models · Hierarchical clustering and kernel density estimation · Statistical classifiers and discriminant analysis · Logistic regression · Support vector machines · High-dimensional analysis · Random intercepts
- Questions that use it (10) — Normal equations for a straight line (answer) · Why a covariance is positive semi-definite (answer) · Whitening and its limits · Normal conditioning and whitening · The fitted LDA model (answer) · A Gram matrix · How many graphs? (answer) · Read a Gaussian graphical model · Q41 (answer) · Question 5
- All 8 questions on Covariance and the multivariate normal
See also
Mean vector and covariance matrix · Variance · A finite stationary point is a global maximiser · Cholesky decomposition · Fisher’s direction · Gram matrix · Kernel density estimator · Least-squares decomposition