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The operator is said to be positive-definite, and written >, if , >, for all {}. [ 1 ] Many authors define a positive operator A {\displaystyle A} to be a self-adjoint (or at least symmetric) non-negative operator.
The positive definiteness of a Hermitian covariance matrix ensures the well-definedness of multivariate distributions. [3] Hermitian matrices are applied in the design and analysis of communications system, especially in the field of multiple-input multiple-output (MIMO) systems. Channel matrices in MIMO systems often exhibit Hermitian properties.
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. The subtle difference between the two is generally overlooked.
Normal operators are important because the spectral theorem holds for them. The class of normal operators is well understood. Examples of normal operators are unitary operators: N* = N −1; Hermitian operators (i.e., self-adjoint operators): N* = N; skew-Hermitian operators: N* = −N; positive operators: N = MM* for some M (so N is self-adjoint).
If A is Hermitian and Ax, x ≥ 0 for every x, then A is called 'nonnegative', written A ≥ 0; if equality holds only when x = 0, then A is called 'positive'. The set of self adjoint operators admits a partial order, in which A ≥ B if A − B ≥ 0. If A has the form B*B for some B, then A is nonnegative; if B is invertible, then A is positive.
In finite dimensions where operators can be represented by matrices, the Hermitian adjoint is given by the conjugate transpose (also known as the Hermitian transpose). The above definition of an adjoint operator extends verbatim to bounded linear operators on Hilbert spaces H {\displaystyle H} .
Each positive element of a C*-algebra is self-adjoint. [3]For each element of a *-algebra, the elements and are self-adjoint, since * is an involutive antiautomorphism. [4] ...
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 ...