The matrix is positive definite if and only if the bilinear form is positive definite and similarly for a positive definite sesquilinear form in the complex case this is a coordinate realization of an inner product on a vector space. Where is the transpose is positive definite johnson 1970 confusingly the discussion of positive definite matrices is often restricted to only hermitian matrices or symmetric matrices in the case of real matrices pease 1965 johnson 1970 marcus and minc 1988 p 182 marcus and minc 1992 p . The thing about positive definite matrices is xtax is always positive for any non zerovector x not just for an eigenvector2 in fact this is an equivalent definition of a matrix being positive definite. A positive definite matrix has positive eigenvalues positive pivots positive determinants and positive energy license creative commons by nc sa. The matrix a is positive definite if all its principal minors have strictly positive determinants if these determinants are nonzero and alternate in signs starting with det then the matrix a is negative definite
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