Sylvester’s criterion
Topic of Covariance and the multivariate normal, part of Statistical learning
Sylvester’s criterion
| Type | Topic |
| Taught in | Covariance and the multivariate normal |
| Part | Statistical learning |
| Questions using it | 2 |
A symmetric matrix is positive definite exactly when all its leading principal minors are positive. Positive semi-definiteness needs every principal minor non-negative, not only the leading ones.
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
- Questions that use it (2) — Why a covariance is positive semi-definite · A Gram matrix (answer)
- All 8 questions on Covariance and the multivariate normal
See also
Positive definite matrix · Joint distribution and density · Mean vector and covariance matrix · Correlation · Multivariate normal distribution · Squared Mahalanobis distance · Linear images of a normal vector · Conditional distribution of a multivariate normal