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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]
Thus, an matrix of complex numbers could be well represented by a matrix of real numbers. The conjugate transpose, therefore, arises very naturally as the result of simply transposing such a matrix—when viewed back again as an n × m {\displaystyle n\times m} matrix made up of complex numbers.
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)}
Also, here, for a positive semidefinite matrix , denotes a positive semidefinite matrix such that =. Note that B {\displaystyle B} is a unique matrix so defined. A generalized version of concurrence for multiparticle pure states in arbitrary dimensions [ 5 ] [ 6 ] (including the case of continuous-variables in infinite dimensions [ 7 ] ) is ...
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
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 ...
The sum of EP matrices A i is an EP matrix if the null-space of the sum is contained in the null-space of each matrix A i. [6] To be an EP matrix is a necessary condition for normality: A is normal if and only if A is EP matrix and AA*A 2 = A 2 A*A. [4] When A is an EP matrix, the Moore-Penrose inverse of A is equal to the group inverse of A. [4]
Above, we pointed out that reducing a Hermitian matrix to tridiagonal form takes flops. This dwarfs the running time of the divide-and-conquer part, and at this point it is not clear what advantage the divide-and-conquer algorithm offers over the QR algorithm (which also takes Θ ( m 2 ) {\displaystyle \Theta (m^{2})} flops for tridiagonal ...