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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]
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 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 ).
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, 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.
In mathematics, an EP matrix (or range-Hermitian matrix [1] or RPN matrix [2]) is a square matrix A whose range is equal to the range of its conjugate transpose A*. Another equivalent characterization of EP matrices is that the range of A is orthogonal to the nullspace of A. Thus, EP matrices are also known as RPN (Range Perpendicular to ...
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 ...
These matrices are traceless, Hermitian, and obey the extra trace orthonormality relation, so they can generate unitary matrix group elements of SU(3) through exponentiation. [1] These properties were chosen by Gell-Mann because they then naturally generalize the Pauli matrices for SU(2) to SU(3), which formed the basis for Gell-Mann's quark ...