In linear algebra, a circulant matrix is a square matrix in which all rows are composed of the same elements and each row is rotated one element to the right relative to the preceding row. It is a particular kind of Toeplitz matrix.
In numerical analysis, circulant matrices are important because they are diagonalized by a discrete Fourier transform, and hence linear equations that contain them may be quickly solved using a fast Fourier transform.[1] They can be interpreted analytically as the integral kernel of a convolution operator on the cyclic group and hence frequently appear in formal descriptions of spatially invariant linear operations. This property is also critical in modern software defined radios, which utilize Orthogonal Frequency Division Multiplexing to spread the symbols (bits) using a cyclic prefix. This enables the channel to be represented by a circulant matrix, simplifying channel equalization in the frequency domain.
In cryptography, a circulant matrix is used in the MixColumns step of the Advanced Encryption Standard.
^Davis, Philip J., Circulant Matrices, Wiley, New York, 1970 ISBN 0471057711
In linear algebra, a circulantmatrix is a square matrix in which all rows are composed of the same elements and each row is rotated one element to the...
cyclic permutation of its vertices. The graph has an adjacency matrix that is a circulantmatrix. The n vertices of the graph can be numbered from 0 to n −...
n × n Hadamard matrix is that n be a square number. A circulantmatrix is manifestly regular, and therefore a circulant Hadamard matrix would have to be...
Multiplication operator Tridiagonal matrix Toeplitz matrix Toral Lie algebra Circulantmatrix Proof: given the elementary matrix e i j {\displaystyle e_{ij}}...
1.1 of. Circulantmatrix, a square Toeplitz matrix with the additional property that a i = a i + n {\displaystyle a_{i}=a_{i+n}} Hankel matrix, an "upside...
roots of unity, the companion matrix and its transpose both reduce to Sylvester's cyclic shift matrix, a circulantmatrix. Consider a polynomial p ( x...
matrices used in mathematics, science and engineering. A matrix (plural matrices, or less commonly matrixes) is a rectangular array of numbers called entries...
1&0&1&0&0&-\end{pmatrix}}} Which is circulant, i.e. each row is a cyclic shift of the previous row. Such a matrix is called a C W ( n , k ) {\displaystyle...
kernel. Bateman transform Convolution kernel Circular convolution Circulantmatrix Differential equations Kernel method List of transforms List of operators...
the lepton masses are given by the squares of the eigenvalues of a circulantmatrix with real eigenvalues, corresponding to the relation m n = μ [ 1 +...
and h are ≤ N, it is reducible to matrix multiplication where the kernel of the integral transform is a circulantmatrix. A case of great practical interest...
accordingly. Clifford algebra Generalizations of Pauli matrices DFT matrixCirculantmatrix Weyl, H. (1927). "Quantenmechanik und Gruppentheorie". Zeitschrift...
fractional integral and fractional derivative. Analog signal processing Circulantmatrix Convolution for optical broad-beam responses in scattering media Convolution...
square matrix. The determinant of a matrix A is commonly denoted det(A), det A, or |A|. Its value characterizes some properties of the matrix and the...
convolution or wrapped convolution. It results from multiplication of a skew circulantmatrix, generated by vector a, with vector b. Circular convolution theorem...
is a symmetric matrix consisting of nine 3 × 3 circulant blocks. Paley Construction II produces the symmetric 20 × 20 Hadamard matrix, 1- 111111 111111...
consequence of the circular convolution theorem is that the DFT matrix F diagonalizes any circulantmatrix. A useful property of the DFT is that the inverse DFT...
linear complexity test implemented in the TestU01 suite; a boolean circulantmatrix initialized from consecutive bits of an LFSR will never have rank greater...
group. The roots of unity appear as entries of the eigenvectors of any circulantmatrix; that is, matrices that are invariant under cyclic shifts, a fact that...
m ≈ log q {\displaystyle m\approx \log q} . Definition: The nega-circulantmatrix of b {\displaystyle b} is defined as: for b = ∑ i = 0 n − 1 b i x i...
definition of a circulantmatrix, and ΛH is a diagonal matrix whose diagonal elements correspond to the first column of the circulant channel matrix H. The receiver...
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