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In mathematics, a Hermitian matrix (or self-adjoint matrix) is a complex square matrix that is equal to its own conjugate transpose —that is, the element in the i -th row and j -th column is equal to the complex conjugate of the element in the j -th row and i -th column, for all indices i and j: or in matrix form:
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: M itself. In other words, all of the leading principal minors must be positive. By using appropriate permutations of rows and columns of M, it can also be shown that the positivity of any ...
Let A be an m × n matrix and k an integer with 0 < k ≤ m, and k ≤ n.A k × k minor of A, also called minor determinant of order k of A or, if m = n, (n−k)th minor determinant of A (the word "determinant" is often omitted, and the word "degree" is sometimes used instead of "order") is the determinant of a k × k matrix obtained from A by deleting m−k rows and n−k columns.
Gell-Mann matrices. The Gell-Mann matrices, developed by Murray Gell-Mann, are a set of eight linearly independent 3×3 traceless Hermitian matrices used in the study of the strong interaction in particle physics. They span the Lie algebra of the SU (3) group in the defining representation.
Conjugate transpose. In mathematics, the conjugate transpose, also known as the Hermitian transpose, of an complex matrix is an matrix obtained by transposing and applying complex conjugation to each entry (the complex conjugate of being , for real numbers and ). There are several notations, such as or , [1] , [2] or (often in physics) .
In linear algebra, the Cholesky decomposition or Cholesky factorization (pronounced / ʃəˈlɛski / 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, the Rayleigh quotient[1] (/ ˈreɪ.li /) for a given complex Hermitian matrix and nonzero vector is defined as: [2][3] For real matrices and vectors, the condition of being Hermitian reduces to that of being symmetric, and the conjugate transpose to the usual transpose . Note that for any non-zero scalar .
In mathematics, Choi's theorem on completely positive maps is a result that classifies completely positive maps between finite-dimensional (matrix) C*-algebras. An infinite-dimensional algebraic generalization of Choi's theorem is known as Belavkin 's "Radon–Nikodym" theorem for completely positive maps.