Definite matrix
Positive semi-definite matrices are defined similarly, except that the scalarsNegative-definite and negative semi-definite matrices are defined analogously.Positive-definite and positive-semidefinite matrices can be characterized in many ways, which may explain the importance of the concept in various parts of mathematics.Positive-definite and positive-semidefinite real matrices are at the basis of convex optimization, since, given a function of several real variables that is twice differentiable, then if its Hessian matrix (matrix of its second partial derivatives) is positive-definite at a pointThe set of positive definite matrices is an open convex cone, while the set of positive semi-definite matrices is a closed convex cone.Notice that this is always a real number for any Hermitian square matrixis real and positive for every non-zero complex column vectorsHermitian matrix (this includes real symmetric matrices).is a unitary complex matrix whose columns comprise an orthonormal basis of eigenvectors ofSince the spectral theorem guarantees all eigenvalues of a Hermitian matrix to be real, the positivity of eigenvalues can be checked using Descartes' rule of alternating signs when the characteristic polynomial of a real, symmetric matrixis positive semidefinite, the eigenvalues are non-negative real numbers, so one can defineas the diagonal matrix whose entries are non-negative square roots of eigenvalues.It is positive definite if and only if it is the Gram matrix of some linearly independent vectors.This statement has an intuitive geometric interpretation in the real case: let the columns ofis positive definite if and only if its quadratic form is a strictly convex function.case, we have Theorem: This quadratic function is strictly convex, and hence has a unique finite global minimum, if and only ifis a line or a downward parabola, thus not strictly convex and not having a global minimum.For this reason, positive definite matrices play an important role in optimization problems.This is so although simultaneous diagonalization is not necessarily performed with a similarity transformation.but note that this is no longer an orthogonal diagonalization with respect to the inner product whereThis defines a partial ordering on the set of all square matrices.[15] Regarding the Hadamard product of two positive semidefinite matricesThe set of positive semidefinite symmetric matrices is convex.This property guarantees that semidefinite programming problems converge to a globally optimal solution.Converse results can be proved with stronger conditions on the blocks, for instance, using the Schur complement.if its gradient is zero and its Hessian (the matrix of all second derivatives) is positive semi-definite at that point.Similar statements can be made for negative definite and semi-definite matrices.determines whether the matrix is positive definite, and is assessed in the narrower sense above.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 negative is inserted in Fourier's law to reflect the expectation that heat will always flow from hot to cold.