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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 row vector transpose of . [1] 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 ...
For the reverse implication, it suffices to show that if has all non-negative principal minors, then for all t>0, all leading principal minors of the Hermitian matrix + are strictly positive, where is the nxn identity matrix. Indeed, from the positive definite case, we would know that the matrices + are strictly positive definite.
By definition, a positive semi-definite matrix, such as , is Hermitian; therefore f(−x ... A function is negative semi-definite if the inequality is reversed.
A symmetric real matrix A is called positive-definite if the associated quadratic form = has a positive value for every nonzero vector x in . If f ( x ) only yields negative values then A is negative-definite ; if f does produce both negative and positive values then A is indefinite . [ 30 ]
In linear algebra, the Cholesky decomposition or Cholesky factorization (pronounced / ʃ ə ˈ l ɛ s k i / shə-LES-kee) is a decomposition of a Hermitian, positive-definite matrix into the product of a lower triangular matrix and its conjugate transpose, which is useful for efficient numerical solutions, e.g., Monte Carlo simulations.
In operator theory, a branch of mathematics, a positive-definite kernel is a generalization of a positive-definite function or a positive-definite matrix. It was first introduced by James Mercer in the early 20th century, in the context of solving integral operator equations. Since then, positive-definite functions and their various analogues ...
The principal square root of a real positive semidefinite matrix is real. [3] The principal square root of a positive definite matrix is positive definite; more generally, the rank of the principal square root of A is the same as the rank of A. [3] The operation of taking the principal square root is continuous on this set of matrices. [4]
A totally positive matrix has all entries positive, so it is also a positive matrix; and it has all principal minors positive (and positive eigenvalues). A symmetric totally positive matrix is therefore also positive-definite. A totally non-negative matrix is defined similarly, except that all the minors must be non-negative (positive or zero ...