= Positive Definiteness = A matrix is '''positive definite''' if all [[LinearAlgebra/EigenvaluesAndEigenvectors|eigenvalues]] are positive. A matrix is '''positive semi-definite''' if all eigenvalues are positive ''or'' zero. <> ---- == Description == A positive definite matrix, by definition, is a [[LinearAlgebra/SpecialMatrices#Symmetric_Matrices|symmetric]] matrix whose [[LinearAlgebra/EigenvaluesAndEigenvectors|eigenvalues]] are all positive. Such a matrix has several useful properties: * Always [[LinearAlgebra/Invertibility|invertible]] * All pivots are positive * The [[LinearAlgebra/Determinant|determinant]] is positive, and all subdeterminants are also positive 'Semi-definite' is a slight modification, allowing 0 as an eigenvalue. === Operations === If '''''A''''' is positive definite, then so is ''c'''A''''' for any real scalar ''c''. If '''''A''''' and '''''B''''' are both positive definite... * so is '''''A''' + '''B'''''. * so is '''''ABA'''''. * so is '''''BAB'''''. If '''''A''''' and '''''B''''' are both positive semi-definite, then so is '''''A''' + '''B'''''. If '''''A''''' is definite and '''''B''''' is semi-definite, then '''''A''' + '''B''''' is definite. ---- CategoryRicottone