We review the properties of the Kronecker (direct, or tensor) product of square matrices A ? B ? C ··· in terms of Hubbard operators. In its simplest form, a Hubbard operator X_n~(i,j) can be expressed as the n-square matrix which has entry 1 in position (i, j) and zero ...
Higher order Hadamard matrices are obtained by \({H}_{{2}^{n+1}}={H}_{{2}^{n}}\otimes {H}_{2}\), where \(\otimes \) is the Kronecker product operator. The i th row/column in the Hadamard matrix can be reshaped to form a square pattern P i . These mathematical operatio...
Harold V. Henderson, Friedrich Pukelsheim, and Shayle R. Searle,On the History of the Kronecker Product, Linear and Multilinear Algebra 14(2), 113–120, 1983. Harold V. Henderson and Shayle R. Searle,The Vec-Permutation Matrix, the Vec Operator and Kronecker Products: A Review, Linear and...
n=0 ∞∞ sin(A) ≡ cos(A) ≡ (375) (376) 10.2 10.2.1 Kronecker and Vec Operator The Kronecker Product The Kronecker product of an m × n matrix A and an r × q matrix B, is an mr × nq matrix, A B dened as A11 B A12 B ... A1n B A21 B A22 B ... A2n B AB=...
This paper studies the properties of the Kronecker product related to the mixed matrix products, the vector operator, and the vec-permutation matrix and gives several theorems and their proofs. In addition, we establish the relations between the singular values of two matrices and their Kronecker ...
(orBS) is decomposed into the standard deviation matrices and horizontal or vertical correlation matrices. Then the NMC method57is used to estimate the standard deviation matrices and the Kronecker product58is employed to decompose the 3-D matrices of horizontal and vertical correlation into 1-D ...
Kronecker Tensor Product The Kronecker product,kron(X,Y), of two matrices is the larger matrix formed from all possible products of the elements ofXwith those ofY. IfXism-by-nandYisp-by-q, thenkron(X,Y)ismp-by-nq. The elements are arranged such that each element ofXis multiplied by the...
The different versions of the fast Fourier transform (FFT) are described here for arbitrary base in terms of the matrix factors of the discrete Fourier transform matrixT_{N}. The Kronecker product notation and the ideal shuffle baserpermutation operator form the basis for a unifying theory through...
The purpose of this paper is to present a matrix inequality on the Kronecker product that unifies the proofs of many existing matrix inequalities in the Löwner partial ordering on the sum, ordinary product, and Hadamard (Schur) product. Schur complements serve as the basic tool. 关键词: Cor...
whereH2is the 2 × 2 mother WH matrix defined as\({{{\bf{H}}}^{2}=\left[\begin{array}{ll}1&1\\ 1&-1\end{array}\right]\)and ⊗ denotes the Kronecker product. The WH matrix is a symmetric unitary matrix; in other words, (1/2d)HH = I. Each of the 2dcolu...