positive definite matrices

(1.8 hours to learn)


A symmetric matrix A is positive definite if x^T A x > 0 for any nonzero vector x, or positive semidefinite if the inequality is not necessarily strict. They can be equivalently characterized in terms of all of the eigenvalues being positive, or all of the pivots in Gaussian elimination being positive. Examples of PSD matrices include covariance matrices and Hessian matrices of convex functions. The singular value decomposition (SVD) is closely related to the eigendecomposition of a positive semidefinite matrix.


This concept has the prerequisites:


  • Know the definition of a positive (semi-)definite matrix (in terms of the quadratic form)
  • Show that a symmetric matrix is PD if and only if its eigenvalues are all positive

Core resources (read/watch one of the following)


MIT Open Courseware: Linear Algebra (2011)
Videos for an introductory linear algebra course focusing on numerical methods.
Author: Gilbert Strang
Additional dependencies:
  • Gaussian elimination


See also