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The Hermitian Laplacian matrix is a key tool in this context, as it is used to analyze the spectra of mixed graphs. [4] The Hermitian-adjacency matrix of a mixed graph is another important concept, as it is a Hermitian matrix that plays a role in studying the energies of mixed graphs. [5]
Other names for the conjugate transpose of a matrix are Hermitian transpose, Hermitian conjugate, adjoint matrix or transjugate. The conjugate transpose of a matrix A {\displaystyle \mathbf {A} } can be denoted by any of these symbols:
Anti-Hermitian matrix: Synonym for skew-Hermitian matrix. Anti-symmetric matrix: Synonym for skew-symmetric matrix. Arrowhead matrix: A square matrix containing zeros in all entries except for the first row, first column, and main diagonal. Band matrix: A square matrix whose non-zero entries are confined to a diagonal band. Bidiagonal matrix
In mathematical analysis, a Hermitian function is a complex function with the property that its complex conjugate is equal to the original function with the variable changed in sign: f ∗ ( x ) = f ( − x ) {\displaystyle f^{*}(x)=f(-x)}
It is the distribution of times the sample Hermitian covariance matrix of zero-mean independent Gaussian random variables. It has support for Hermitian positive definite matrices. [1] The complex Wishart distribution is the density of a complex-valued sample covariance matrix. Let
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.
Moreover, the following completeness identity for the above Hermite functions holds in the sense of distributions: = () = (), where δ is the Dirac delta function, ψ n the Hermite functions, and δ(x − y) represents the Lebesgue measure on the line y = x in R 2, normalized so that its projection on the horizontal axis is the usual Lebesgue ...
Applicable to: square, hermitian, positive definite matrix Decomposition: =, where is upper triangular with real positive diagonal entries Comment: if the matrix is Hermitian and positive semi-definite, then it has a decomposition of the form = if the diagonal entries of are allowed to be zero