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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 .
In mathematics, positive semidefinite may refer to: Positive semidefinite function; Positive semidefinite matrix; Positive semidefinite quadratic form;
The Gram matrix is symmetric in the case the inner product is real-valued; it is Hermitian in the general, complex case by definition of an inner product. The Gram matrix is positive semidefinite, and every positive semidefinite matrix is the Gramian matrix for some set of vectors. The fact that the Gramian matrix is positive-semidefinite can ...
The class of copositive matrices can be characterized using principal submatrices. One such characterization is due to Wilfred Kaplan: [6]. A real symmetric matrix A is copositive if and only if every principal submatrix B of A has no eigenvector v > 0 with associated eigenvalue λ < 0.
A positive semidefinite matrix A can also have many matrices B such that =. However, A always has precisely one square root B that is both positive semidefinite and symmetric. In particular, since B is required to be symmetric, B = B T {\displaystyle B=B^{\textsf {T}}} , so the two conditions A = B B {\displaystyle A=BB} or A = B T B ...
In mathematics, the polar decomposition of a square real or complex matrix is a factorization of the form =, where is a unitary matrix and is a positive semi-definite Hermitian matrix (is an orthogonal matrix and is a positive semi-definite symmetric matrix in the real case), both square and of the same size.
The trace distance is defined as half of the trace norm of the difference of the matrices: (,):= ‖ ‖ = [() † ()], where ‖ ‖ [†] is the trace norm of , and is the unique positive semidefinite such that = (which is always defined for positive semidefinite ).
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 ...
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