positive definite


The definiteness of a matrix is an important property that has use in many areas of mathematics and physics. Below are some examples:

  1. 1.

    In optimizing problems, the definiteness of the Hessian matrix determines the quality of an extremal value. The full details can be found on this page (http://planetmath.org/RelationsBetweenHessianMatrixAndLocalExtrema).

Definition [1] Suppose A is an n×n square Hermitian matrixMathworldPlanetmath. If, for any non-zero vector x, we have that


then A a positive definitePlanetmathPlanetmath matrix. (Here x=x¯t, where x¯ is the complex conjugateMathworldPlanetmath of x, and xt is the transposeMathworldPlanetmath of x.)

One can show that a Hermitian matrix is positive definite if and only if all its eigenvaluesMathworldPlanetmathPlanetmathPlanetmathPlanetmath are positive [1]. Thus the determinantMathworldPlanetmath of a positive definite matrix is positive, and a positive definite matrix is always invertiblePlanetmathPlanetmath. The Cholesky decompositionMathworldPlanetmath provides an economical method for solving linear equations involving a positive definite matrix. Further conditions and properties for positive definite matrices are given in [2].


  • 1 M. C. Pease, Methods of Matrix Algebra, Academic Press, 1965
  • 2 C.R. Johnson, Positive definite matrices, American Mathematical Monthly, Vol. 77, Issue 3 (March 1970) 259-264.
Title positive definite
Canonical name PositiveDefinite
Date of creation 2013-03-22 12:20:03
Last modified on 2013-03-22 12:20:03
Owner matte (1858)
Last modified by matte (1858)
Numerical id 10
Author matte (1858)
Entry type Definition
Classification msc 15A48
Related topic PositiveSemidefinite
Related topic NegativeDefinite
Related topic QuadraticForm
Related topic EuclideanVectorSpace
Related topic EuclideanVectorSpace2