Definite matrix - Wikipedia
linear-algebramatrix-theoryoptimizationstatistics
Abstraction: Symmetric matrix with all positive eigenvalues defining inner product
Key points:
- An n×n Hermitian matrix M is positive-definite iff z*Mz > 0 for all nonzero z, equivalently iff all eigenvalues are strictly positive.
- Equivalent characterizations: congruent to a diagonal matrix with positive entries; all leading principal minors positive (Sylvester's criterion); admits Cholesky factorization M = LL*.
- M is positive semidefinite iff it is a Gram matrix of some set of vectors; positive definite iff those vectors are linearly independent.
- Positive-definiteness of the Hessian at a point certifies a local minimum for smooth functions; the set of PD matrices is an open convex cone.
- Cholesky decomposition M = LL^T is the computationally efficient factorization for PD matrices; LDL^T variant avoids square roots.
- Covariance matrices of multivariate distributions are always positive semidefinite; positive definite unless one variable is a linear function of others.
Connections: Positive Definite Matrix · Eigenvalues · Cholesky Decomposition · Convex Optimization
Source: http://en.wikipedia.org/wiki/Positive-definite_matrix