Search results
Results from the WOW.Com Content Network
Printable version; In other projects Wikidata item; Appearance. move to sidebar hide. In mathematics, positive semidefinite may refer to: Positive semidefinite ...
In mathematics (specifically linear algebra, operator theory, and functional analysis) as well as physics, a linear operator acting on an inner product space is called positive-semidefinite (or non-negative) if, for every (), , and , , where is the domain of .
unstrict inequality signs (less-than or equals to sign and greater-than or equals to sign) 1670 (with the horizontal bar over the inequality sign, rather than below it) John Wallis: 1734 (with double horizontal bar below the inequality sign) Pierre Bouguer
According to that sign, the quadratic form is called positive-definite or negative-definite. A semidefinite (or semi-definite) quadratic form is defined in much the same way, except that "always positive" and "always negative" are replaced by "never negative" and "never positive", respectively.
In mathematics, the polar decomposition of a square real or complex matrix is a factorization of the form =, where is a unitary matrix and is a positive semi-definite Hermitian matrix (is an orthogonal matrix and is a positive semi-definite symmetric matrix in the real case), both square and of the same size.
Uniqueness: is always unique and equal to (which is always hermitian and positive semidefinite). If A {\displaystyle A} is invertible, then U {\displaystyle U} is unique. Comment: Since any Hermitian matrix admits a spectral decomposition with a unitary matrix, P {\displaystyle P} can be written as P = V D V ∗ {\displaystyle P=VDV^{*}} .
Let denote the space of Hermitian matrices, + denote the set consisting of positive semi-definite Hermitian matrices and + + denote the set of positive definite Hermitian matrices. For operators on an infinite dimensional Hilbert space we require that they be trace class and self-adjoint , in which case similar definitions apply, but we discuss ...
As the transposition map preserves eigenvalues, the spectrum of () is the same as the spectrum of , and in particular () must still be positive semidefinite. Thus ρ T B {\displaystyle \rho ^{T_{B}}} must also be positive semidefinite.