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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 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 ...
in which is the reduced density matrix (or its continuous-variable analogue [7]) across the bipartition of the pure state, and it measures how much the complex amplitudes deviate from the constraints required for tensor separability. The faithful nature of the measure admits necessary and sufficient conditions of separability for pure states.
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 ...
As stated in the introduction, for any vector x, one has (,) [,], where , are respectively the smallest and largest eigenvalues of .This is immediate after observing that the Rayleigh quotient is a weighted average of eigenvalues of M: (,) = = = = where (,) is the -th eigenpair after orthonormalization and = is the th coordinate of x in the eigenbasis.