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Hilbert spaces
Published in John P. D’Angelo, Linear and Complex Analysis for Applications, 2017
Thus a Hermitian matrix can be unitarily diagonalized. As we noted in Chapter 1, the column vectors of U are the eigenvectors of A. The conclusions of Theorem 5.7 and Corollary 5.8 also hold for normal matrices; A is normal if AA* = A* A. The proofs are essentially the same. See, for example, [HK].
Hermitian and Positive Definite Matrices
Published in Darald J. Hartfiel, ®, 2017
As we have seen, a Hermitian can be diagonalized by a unitary matrix. By using a special class of Hermitian matrices, the positive definite Hermitian matrices, we show in this section how two matrices can be simultaneously diagonalized in a special way.
Matrices
Published in Alexander D. Poularikas, Adaptive Filtering, 2017
A matrix is said to be Hankel if its elements along the diagonals that are perpendicular to the main diagonal are equal. A complex matrix is said to be Hermitian if it is conjugate symmetric. () Hankel:A=[a11a12a13a12a13a23a13a23a33]Hermitian:A=[11+2j-j1-2j24jj-4j6]=(A*)T=AH
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.
Modal Identification of a Soil-subway System with Emphasis on Scattering of Seismic Waves Induced by Uniform and Non-uniform Support Excitations
Published in Journal of Earthquake Engineering, 2022
Mohsen Isari, Majid Damadipour, Abbasali Taghavi Ghalesari, Seyyed Kazem Razavi, Reza Tarinejad
where and are unitary matrices containing the left and right singular vectors and (), respectively. Matrix S is a diagonal matrix that includes the scalar singular values (). The superscript is used to show the Hermitian transpose (complex conjugate transpose). The spectral peaks of the first singular values are used to identify the vibration modes, and the first left singular vector corresponding to the desired mode is also used to extract the modal shape of the system (Tarinejad and Damadipour 2014, 2016).
Unidirectional weak visibility in bandgap and singular scattering in conduction band of one-dimensional -symmetry photonic crystal
Published in Waves in Random and Complex Media, 2022
In Figure 3, we present the complex band structures for different values of the imaginary relative permittivity . In our calculation, plane wave expansion method is applied, in our work, 61 plane waves are used to expand the electromagnetic field. For small values of , e.g. in Figure 3(a,b), the eigenfrequencies for all the bands remains real. When is increased to a critical value, the second bandgap starts to vanish and the 2nd and 3rd bands begin to coalesce, in which case an exceptional point emerges at the center of the Brillouin zone. As is further increased, this exceptional point is split in two, e.g. in Figure 3(c,d). These points are located at either sides of the Brillouin center; between these two exceptional points the complex eigenfrequencies emerge, which are conjugate to each other. The band region with complex eigenfrequencies forms the -broken phase, while the band region with real eigenfrequencies corresponds to the -exact phase. These properties agree with non-Hermitian quantum mechanics.