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Kronecker Product Factorization and FFTs
Published in Eleanor Chu, Discrete and Continuous Fourier Transforms, 2008
3. The Kronecker product is distributive with respect to addition, that is, (A+B)⊗C=A⊗C+B⊗C;A⊗(B+C)=A⊗B+A⊗C.
Compressive Sensing in the Multiple Antenna Regime
Published in Ashish Bagwari, Geetam Singh Tomar, Jyotshana Bagwari, Advanced Wireless Sensing Techniques for 5G Networks, 2018
The Kronecker product is denoted by the symbol ⊗. Then the sparsifying basis Ψ associated with 2D-DCT is given as (Figure 7.2); Ψ2D−DCT=(CNt⊗CNR)T
Elements of Quantum Electronics
Published in Michael Olorunfunmi Kolawole, Electronics, 2020
The two products σx⊗σz (Equation 9.41) and σz⊗σx (Equation 9.42) are not the same because they represent different observables. Realistically, tensor product is a special case of the Kronecker product of matrices. It is therefore conceivable that matrices of abstract operators and state vectors can replicate a known behavior. For instance, (σz⊗σx)|uv〉=[01001000000−100−10][0100]=[1000]
Extended dissipativity and dynamical output feedback control for interval type-2 singular semi-Markovian jump fuzzy systems
Published in International Journal of Systems Science, 2022
Notation. Throughout this paper, the real matrix represents P being a positive definite (or positive semi-definite) matrix. The symbols and denote the inverse and the transpose of the matrix P, respectively. refers to the maximum eigenvalue of P. The expression is used to denote . The set , non-negative integer set and non-negative real number set are represented by , and , respectively. and stand for the sets and , respectively. The Kronecker product is denoted by ⊗. The symbol ★ indicates irrelevant items and * means symmetric terms in symmetric block matrices. () is a probability space. represents a mathematical expectation operator.
Second-order consensus in multi-agent systems with nonlinear dynamics and intermittent control
Published in International Journal of Systems Science, 2020
Zeyu Han, Qiang Jia, Wallace K. S. Tang
For a matrix A, denotes its transpose and is its inverse. and are the largest and smallest eigenvalues of A, respectively. is the largest entry of vector ξ. For any vector or matrix, represents the Euclidian norm. The symbol ⊗ denotes Kronecker product. is a column vector of all ones (zeros). is the identity matrix of order N. is an matrix with all zeros.
Tensor Mixed Effects Model With Application to Nanomanufacturing Inspection
Published in Technometrics, 2020
Xiaowei Yue, Jin Gyu Park, Zhiyong Liang, Jianjun Shi
Matricization, also known as unfolding or flattening, is the process of reordering the elements of a tensor into a matrix (Kolda and Bader 2009). The k-mode matricization of a tensor is denoted by . is the vectorization of a tensor . The k-mode product of a tensor with a matrix is denoted by and elementwise, we have , where all the indices range from 1 to their capital versions, for example, the index j goes from , and the index ik goes from . The Kronecker product of matrices and are denoted by . The Kronecker product is an operation on two matrices resulting in a block matrix and it is a generalization of the outer product.