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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.
A more subtle example of the distinction between symmetric and (essentially) self-adjoint operators comes from Schrödinger operators in quantum mechanics. If the potential energy is singular—particularly if the potential is unbounded below—the associated Schrödinger operator may fail to be essentially self-adjoint.
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.
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.
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.
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 ...
Even if is not square, the two matrices and are both Hermitian and in fact positive semi-definite matrices. The conjugate transpose "adjoint" matrix A H {\displaystyle \mathbf {A} ^{\mathrm {H} }} should not be confused with the adjugate , adj ( A ) {\displaystyle \operatorname {adj} (\mathbf {A} )} , which is also sometimes called adjoint .