In mathematics, a symmetric matrix with real entries is positive-definite if the real number is positive for every nonzero real column vector where is the transpose of More generally, a Hermitian matrix (that is, a complex matrix equal to its conjugate transpose) is positive-definite if the real number is positive for every nonzero complex column vector where denotes the conjugate transpose of Positive semi-definite matrices are defined similarly, except that the scalars and are required to be positive or zero (that is, nonnegative). Negative-definite and negative semi-definite matrices are defined analogously. A matrix that is not positive semi-definite and not negative semi-definite is sometimes called indefinite. A matrix is thus positive-definite if and only if it is the matrix of a positive-definite quadratic form or Hermitian form. In other words, a matrix is positive-definite if and only if it defines an inner product. 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. A matrix is positive-definite if and only if it satisfies any of the following equivalent conditions. is congruent with a diagonal matrix with positive real entries. is symmetric or Hermitian, and all its eigenvalues are real and positive. is symmetric or Hermitian, and all its leading principal minors are positive. There exists an invertible matrix with conjugate transpose such that A matrix is positive semi-definite if it satisfies similar equivalent conditions where "positive" is replaced by "nonnegative", "invertible matrix" is replaced by "matrix", and the word "leading" is removed. 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 point , then the function is convex near , and, conversely, if the function is convex near , then the Hessian matrix is positive-semidefinite at . Some authors use more general definitions of definiteness, including some non-symmetric real matrices, or non-Hermitian complex ones. Definitions In the following definitions, is the transpose of , is the conjugate transpose of and denotes the n-dimensional zero-vector. Definitions for real matrices An symmetric real matrix is said to be positive-definite if for all non-zero in . Formally, An symmetric real matrix is said to be positive-semidefinite or non-negative-definite if for all in . Formally, An symmetric real matrix is said