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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 fact that any Hermitian complex 2 × 2 matrices can be expressed in terms of the identity matrix and the Pauli matrices also leads to the Bloch sphere representation of 2 × 2 mixed states’ density matrix, (positive semidefinite 2 × 2 matrices with unit trace.
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 ...
The general form of an inner product on is known as the Hermitian form and is given by , = † = † ¯, where is any Hermitian positive-definite matrix and † is the conjugate transpose of . For the real case, this corresponds to the dot product of the results of directionally-different scaling of the two vectors, with positive scale factors ...
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 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 ...
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.