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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 .
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 .
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 ...
The diagonal elements must be real, as they must be their own complex conjugate. Well-known families of Hermitian matrices include the Pauli matrices, the Gell-Mann matrices and their generalizations. In theoretical physics such Hermitian matrices are often multiplied by imaginary coefficients, [6] [7] which results in skew-Hermitian matrices.
In linear algebra, the Cholesky decomposition or Cholesky factorization (pronounced / ʃ ə ˈ l ɛ s k i / shə-LES-kee) is a decomposition of a Hermitian, positive-definite matrix into the product of a lower triangular matrix and its conjugate transpose, which is useful for efficient numerical solutions, e.g., Monte Carlo simulations.
In mathematics, specifically in operator theory, each linear operator on an inner product space defines a Hermitian adjoint (or adjoint) operator on that space according to the rule A x , y = x , A ∗ y , {\displaystyle \langle Ax,y\rangle =\langle x,A^{*}y\rangle ,}
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).
In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space [1] [2]) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar , often denoted with angle brackets such as in a , b {\displaystyle \langle a,b\rangle } .