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Harmonic Analysis on the Euclidean Motion Groups
Published in Gregory S. Chirikjian, Alexander B. Kyatkin, Engineering Applications of Noncommutative Harmonic Analysis, 2021
Gregory S. Chirikjian, Alexander B. Kyatkin
i.e., ℱ(f*)=(f^)†is the complex conjugate transpose of the Fourier transform matrix f^. This means that h(e)=∫0∞trace(f^†(p)f^(p))pdp=∫0∞‖f^(p)‖22pdp.
Review of Linear Algebra
Published in Mohammad Monir Uddin, Computational Methods for Approximation of Large-Scale Dynamical Systems, 2019
Transpose and conjugate transpose of a matrix: The transpose of A ∈ ℝm×n is another n × m matrix AT obtained transpose matrix by interchanging the rows and columns in A. On the other hand, the conjugate transpose or transpose conjugateHermitian transpose of Hermitian transposeA ∈ ℂm×n is A∗∈Cn×mHermitian obtained from A by taking the transpose and then taking the complex conjugate of each of the entries. complex conjugate
Force-System Resultants and Equilibrium
Published in Richard C. Dorf, The Engineering Handbook, 2018
usually denoted by AT. A matrix A such that A=AT is said to be symmetric. The conjugate transpose of A is the matrix that results from replacing each element of AT by its complex conjugate, and is usually denoted by AH. A matrix such that A=AH is said to be Hermitian.
Generalised KYP lemma with its application in finite frequency H ∞ distributed filter design for nonideally interconnected networked control systems
Published in International Journal of Systems Science, 2021
Xuefeng Chen, Huiling Xu, Xiaokai Zhai, Zhiping Lin
Notations: The set of real numbers and nonnegative real numbers are denoted by and , respectively. and denote n-dimensional real and complex space, respectively. , , and refer to the real matrix set, real symmetric matrix set, complex matrix set and Hermitian matrix set, respectively. The identity and zero matrix are denoted by and , respectively, or just I and 0. For a matrix , its complex conjugate transpose is denoted by . indicates . The image space of a matrix A is denoted by . means that A is a negative definite (negative semi-definite) matrix. denotes the maximum singular value of matrix A. The notation ★ is used when the terms can be induced by symmetry. is used to denote the vector . The block-diagonal matrix is denoted by , while denotes the stacked vector/matrix.
MIMO Nyquist interpretation of the large gain theorem
Published in International Journal of Control, 2020
Ryan James Caverly, Richard Pates, Leila Jasmine Bridgeman, James Richard Forbes
In this paper, boldface letters represent matrices, script letters denote operators, and simple letters denote scalars. The identify matrix is written as and a matrix filled with zeros is written as . Summation points within block diagrams are positive unless otherwise noted. Recall that if , and if , , where for and for t>T. The minimum and maximum singular values of a matrix are and , respectively, where are the singular values of . Using a slight abuse of notation, an by matrix of proper real rational transfer functions is denoted as . The norm of an asymptotically stable transfer matrix is . The complex conjugate transpose of the matrix is . The number of counter-clockwise (CCW) encirclements of the origin made by the Nyquist plot of , also known as the winding number about the origin, is denoted as . The number of closed right-half plane (CRHP) poles of counted according to the Smith-McMillan degree (Skogestad & Postlethwaite, 2007, p. 154) is written as .