made for negative definite and semi-definite matrices. Covariance In statistics, the covariance matrix of a multivariate probability distribution is always positive semi-definite; and it is positive definite unless one variable is an exact linear function of the others. Conversely, every positive semi-definite matrix is the covariance matrix of some multivariate distribution. Extension for non-Hermitian square matrices The definition of positive definite can be generalized by designating any complex matrix (e.g. real non-symmetric) as positive definite if for all non-zero complex vectors , where denotes the real part of a complex number . Only the Hermitian part determines whether the matrix is positive definite, and is assessed in the narrower sense above. Similarly, if and are real, we have for all real nonzero vectors if and only if the symmetric part is positive definite in the narrower sense. It is immediately clear that is insensitive to transposition of M. Consequently, a non-symmetric real matrix with only positive eigenvalues does not need to be positive definite. For example, the matrix has positive eigenvalues yet is not positive definite; in particular a negative value of is obtained with the choice (which is the eigenvector associated with the negative eigenvalue of the symmetric part of In summary, the distinguishing feature between the real and complex case is that, a bounded positive operator on a complex Hilbert space is necessarily Hermitian, or self adjoint. The general claim can be argued using the polarization identity. That is no longer true in the real case. Applications Heat conductivity matrix Fourier's law of heat conduction, giving heat flux in terms of the temperature gradient is written for anisotropic media as , in which is the symmetric thermal conductivity matrix. The negative is inserted in Fourier's law to reflect the expectation that heat will always flow from hot to cold. In other words, since the temperature gradient always points from cold to hot, the heat flux is expected to have a negative inner product with so that . Substituting Fourier's law then gives this expectation as , implying that the conductivity matrix should be positive definite. See also Covariance matrix M-matrix Positive-definite function Positive-definite kernel Schur complement Sylvester's criterion Numerical range Notes References External links Wolfram MathWorld: Positive Definite Matrix Matrices de:Definitheit#Definitheit von Matrizen