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Download as PDF; Printable version; ... In mathematics, positive semidefinite may refer to: Positive semidefinite function ... at 17:28 (UTC). Text is ...
A result of Albrecht Pfister [8] shows that a positive semidefinite form in n variables can be expressed as a sum of 2 n squares. [9] Dubois showed in 1967 that the answer is negative in general for ordered fields. [10] In this case one can say that a positive polynomial is a sum of weighted squares of rational functions with positive ...
In contrast with the complex case, a positive-semidefinite operator on a real Hilbert space may not be symmetric. As a counterexample, define A : R 2 → R 2 {\displaystyle A:\mathbb {R} ^{2}\to \mathbb {R} ^{2}} to be an operator of rotation by an acute angle φ ∈ ( − π / 2 , π / 2 ) . {\displaystyle \varphi \in (-\pi /2,\pi /2).}
The fidelity between two quantum states and , expressed as density matrices, is commonly defined as: [1] [2] (,) = ().The square roots in this expression are well-defined because both and are positive semidefinite matrices, and the square root of a positive semidefinite matrix is defined via the spectral theorem.
A spectrahedron. In convex geometry, a spectrahedron is a shape that can be represented as a linear matrix inequality.Alternatively, the set of n × n positive semidefinite matrices forms a convex cone in R n × n, and a spectrahedron is a shape that can be formed by intersecting this cone with an affine subspace.
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 ...
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In this polar decomposition, the unit circle has been replaced by the line x = 1, the polar angle by the slope y/x, and the radius x is negative in the left half-plane. If x 2 ≠ y 2, then the unit hyperbola x 2 − y 2 = 1 and its conjugate x 2 − y 2 = −1 can be used to form a polar decomposition based on the branch of the unit hyperbola ...