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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 new operator can be used to create an object wrapper for a Boolean primitive. However, the typeof operator does not return boolean for the object wrapper, it returns object. Because all objects evaluate as true, a method such as .valueOf(), or .toString(), must be used to retrieve the wrapped value.
If A is an invertible operator on a finite-dimensional Hilbert space, then there is a unique unitary operator U and positive operator P such that =; this is the polar decomposition of A. The positive operator P is the unique positive square root of the positive operator A ∗ A, and U is defined by U = AP −1.
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 mathematics, Sylvester’s criterion is a necessary and sufficient criterion to determine whether a Hermitian matrix is positive-definite. Sylvester's criterion states that a n × n Hermitian matrix M is positive-definite if and only if all the following matrices have a positive determinant: the upper left 1-by-1 corner of M,
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).
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.
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.