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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:
The following tables list several unitary quantum logic gates, together with their common name, how they are represented, and some of their properties. Controlled or conjugate transpose ( adjoint ) versions of some of these gates may not be listed.
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 ...
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 ...
In mathematics, the Rayleigh quotient [1] (/ ˈ r eɪ. l i /) 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 x ∗ {\displaystyle x^{*}} to the usual transpose x ...
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)}
A disadvantage is that compact schemes are implicit and require to solve a diagonal matrix system for the evaluation of interpolations or derivatives at all grid points. Due to their excellent stability properties, compact schemes are a popular choice for use in higher-order numerical solvers for the Navier-Stokes Equations .