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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 .
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 ...
[citation needed] According to the spectral theorem, the continuous functional calculus can be applied to obtain an operator T 1/2 such that T 1/2 is itself positive and (T 1/2) 2 = T. The operator T 1/2 is the unique non-negative square root of T. [citation needed] A bounded non-negative operator on a complex Hilbert space is self adjoint by ...
Positive-definiteness arises naturally in the theory of the Fourier transform; it can be seen directly that to be positive-definite it is sufficient for f to be the Fourier transform of a function g on the real line with g(y) ≥ 0.
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 ...
The diagonal elements must be real, as they must be their own complex conjugate. Well-known families of Hermitian matrices include the Pauli matrices, the Gell-Mann matrices and their generalizations. In theoretical physics such Hermitian matrices are often multiplied by imaginary coefficients, [6] [7] which results in skew-Hermitian matrices.
In practical terms, having an essentially self-adjoint operator is almost as good as having a self-adjoint operator, since we merely need to take the closure to obtain a self-adjoint operator. In physics, the term Hermitian refers to symmetric as well as self-adjoint operators alike.
A form is called strongly positive if it is a linear combination of products of semi-positive forms, with positive real coefficients. A real (p, p) -form η {\displaystyle \eta } on an n -dimensional complex manifold M is called weakly positive if for all strongly positive (n-p, n-p) -forms ζ with compact support, we have ∫ M η ∧ ζ ≥ 0 ...