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Review of Linear Algebra
Published in Mohammad Monir Uddin, Computational Methods for Approximation of Large-Scale Dynamical Systems, 2019
Symmetric matrix and Hermitian matrix: A real matrix A ∈ ℝn×n is called symmetric if A = AT. If A is symmetric, then A − AT = 0 (zero matrix). Hermitian matrix On the other hand, a complex matrix A ∈ ℂn×n is called Hermitian symmetric matrix matrix (or self-adjoint matrixself-adjoint matrix) if it is equal to its own conjugate transpose, that is the element in the i-th row and j-th column is equal to the complex conjugate of the element in the j-th row and i-th column, for all indices i and j.
Jones Matrices and Polarization Properties
Published in Russell A. Chipman, Wai-Sze Tiffany Lam, Garam Young, Polarized Light and Optical Systems, 2018
Russell A. Chipman, Wai-Sze Tiffany Lam, Garam Young
The eigenvalues of a Hermitian matrix are real. Thus, Hermitian matrices represent measurements, just as a polarizer can be inserted in a beam with a radiometer to detect the power in a particular polarization state. The eigenvectors of Hermitian matrices are orthogonal. Hermitian matrices are stretching and compressing matrices and will deform a circle or sphere of unit vectors into an ellipsoid of vectors (which may have complex elements). The maximum and minimum length vectors generated are the eigenvectors and have lengths scaled by the eigenvalues.
Quantum Preprocessing for Deep Convolutional Neural Networks in Atherosclerosis Detection
Published in Siddhartha Bhattacharyya, Mario Köppen, Elizabeth Behrman, Ivan Cruz-Aceves, Hybrid Quantum Metaheuristics, 2022
Siddhartha Bhattacharyya, Mario Köppen, Elizabeth Behrman, Ivan Cruz-Aceves
A measurement must extract the information of a qubit, yielding a numerical result. If a quantum state |ψ〉 is measured, then only one classical state |j〉 can be seen with probability |αj|2, which corresponds to the amplitude αj. This is known as Born's rule. Accordingly, observing a quantum state induces a measurement in the computational basis state probabilities. However, other kinds of measurements can be computed from quantum systems, such as the expectation of an observable and its variance. Observables are Hermitian matrix M such that their eigenvalues are real numbers. The Pauli gates (σx, σy, and σz) are examples of observables, with eigenvalues ±1. Ergo, the expected value of an observable M on a state |ψ〉 is 〈M〉ψ=〈ψ|M|ψ〉.
Stochastic model updating for assembled structures with bolted joints using a Bayesian method
Published in Engineering Optimization, 2022
Yong Zhang, Yan Zhao, Huajiang Ouyang
The excitation (t) in Equation (5) is a stationary random excitation and its PSD is denoted by . Since is a Hermitian matrix, it can be decomposed as where superscript H represents the complex conjugate transpose; and are the eigen-pairs of the Hermitian matrix, which satisfy the following relationships: According to the PEM, the following pseudo-excitations can be constructed: where ‘*’ represents the complex conjugate.
Quasirelativistic two-component core excitations and polarisabilities from a damped-response formulation of the Bethe–Salpeter equation
Published in Molecular Physics, 2020
Max Kehry, Yannick J. Franzke, Christof Holzer, Wim Klopper
As shown by the authors of Ref. [5], the framework of solving the quasirelativistic two-component Bethe–Salpeter equation [43, 44] is formally related to that of two-component TD-DFT [41, 45–47]. Similar to the zeroth-order regular-approximation (ZORA) based ansatz for dynamic polarisabilities using complex frequencies by Devarajan et al. [23], we start from the response equation (in atomic units) where the hermitian matrix and the complex symmetric matrix collect the matrix elements of the electronic Hessian. ω is a complex frequency at which a desired polarisability shall be evaluated and contains the electric dipole integrals transformed into the spinor basis. Note that the linear matrix pencil on the left hand side of Equation (1) has singularities at the (real) excitation energies. Close to these singularities, the imaginary part of the frequency ω that is inserted must not vanish if a stable set of equations is to be obtained.
Sampled-data state-estimation of delayed complex-valued neural networks
Published in International Journal of Systems Science, 2020
Nallappan Gunasekaran, Guisheng Zhai
Notations. and denote the set of n-dimensional complex vectors, complex matrices, respectively. The superscript T and denotes the matrix transposition and complex conjugate transpose, respectively; i denotes the imaginary unit, that is . The asterisk in Hermitian matrix is defined as the conjugate transpose of block. For any matrix P, P>0 means P is positive Hermitian matrix. For complex number z = x + iy, the notation stands for the module of z and ; stands for diagonal of the block- diagonal matrix. If , denotes by its operator norm, i.e..