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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 .
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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.
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.
Semidefinite programming (SDP) is a subfield of mathematical programming concerned with the optimization of a linear objective function (a user-specified function that the user wants to minimize or maximize) over the intersection of the cone of positive semidefinite matrices with an affine space, i.e., a spectrahedron. [1]
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^{*}} .
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 ...
In English, negation is achieved by adding not after the verb. As a practical matter, Modern English typically uses a copula verb (a form of be) or an auxiliary verb with not. If no other auxiliary verb is present, then dummy auxiliary do (does, did) is normally introduced – see do-support. For example, (8) a. I have gone (affirmative) b.