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Linear algebra matrix
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
Circulant_matrix
Open conjecture that no real circulant Hadamard matrix has order greater than 4
conjecture on circulant Hadamard matrices, also called the circulant Hadamard matrix conjecture, states that no real circulant Hadamard matrix has order greater
Ryser's conjecture on circulant Hadamard matrices
Ryser's_conjecture_on_circulant_Hadamard_matrices
Undirected graph acted on by a vertex-transitive cyclic group of symmetries
cyclic permutation of its vertices. The graph has an adjacency matrix that is a circulant matrix. The n vertices of the graph can be numbered from 0 to n −
Circulant_graph
Mathematics concept
n × n Hadamard matrix is that n be a square number. A circulant matrix is manifestly regular, and therefore a circulant Hadamard matrix would have to be
Hadamard_matrix
Matrix with shifting rows
Grudsky. Circulant matrix, a square Toeplitz matrix with the additional property that a i = a i + n {\displaystyle a_{i}=a_{i+n}} Hankel matrix, an "upside
Toeplitz_matrix
Matrix equal to its transpose
Skew-symmetric matrix (also called antisymmetric or antimetric) Centrosymmetric matrix Circulant matrix Covariance matrix Coxeter matrix GCD matrix Hankel matrix Hilbert
Symmetric_matrix
Most widely known generalized inverse of a matrix
pseudoinverse trivially coincides with the matrix itself: A + = A . {\displaystyle A^{+}=A.} For a circulant matrix C {\displaystyle C} , the singular value
Moore–Penrose_inverse
Mathematical weight device
&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
Weighing_matrix
Matrix whose only nonzero elements are on its main diagonal
Multiplication operator Tridiagonal matrix Toeplitz matrix Toral Lie algebra Circulant matrix Proof: given the elementary matrix e i j {\displaystyle e_{ij}}
Diagonal_matrix
Square matrix constructed from a monic polynomial
roots of unity, the companion matrix and its transpose both reduce to Sylvester's cyclic shift matrix, a circulant matrix. Consider a polynomial p ( x
Companion_matrix
Hadamard matrix that is also circulant corresponds to a Menon design with a regular cyclic automorphism group. The existence of such circulant examples
Regular_Hadamard_matrix
matrices used in mathematics, science and engineering. A matrix (plural matrices, or less commonly matrixes) is a rectangular array of numbers called entries
List_of_named_matrices
Sequence of digital values used for synchronisation
nonzero shift. The circulant matrix whose first row is the sequence therefore has mutually orthogonal rows and is a circulant Hadamard matrix of order N {\displaystyle
Barker_code
Mathematical operation
and h are ≤ N, it is reducible to matrix multiplication where the kernel of the integral transform is a circulant matrix. A case of great practical interest
Circular_convolution
Model of multi-species population dynamics
their spatial interactions, then the interaction matrix is circulant. The eigenvalues of a circulant matrix are given by λ k = ∑ j = 0 N − 1 c j γ k j {\displaystyle
Competitive Lotka–Volterra equations
Competitive_Lotka–Volterra_equations
Computational problem used in cryptography
m ≈ log q {\displaystyle m\approx \log q} . Definition: The nega-circulant matrix of b {\displaystyle b} is defined as: for b = ∑ i = 0 n − 1 b i x i
Short integer solution problem
Short_integer_solution_problem
Cryptographic hash function
0\leq j<8\\\end{aligned}}} A circulant matrix C ∈ M n × n {\displaystyle C\in {\mathcal {M}}_{n\times n}} is defined as a matrix where each row is a cyclic
Whirlpool_(hash_function)
are indexed by field elements in the usual 0, 1, 2, … order, Q is a circulant matrix. That is, each row is obtained from the row above by cyclic permutation
Paley_construction
Matrix equal to its conjugate-transpose
In mathematics, a Hermitian matrix (or self-adjoint matrix) is a square matrix with complex-valued entries that is equal to its own conjugate transpose
Hermitian_matrix
Mathematical problem
problem, named after Jacques Hadamard, asks for the largest determinant of a matrix with elements equal to 1 or −1. The analogous question for matrices with
Hadamard's maximal determinant problem
Hadamard's_maximal_determinant_problem
determined accordingly. Generalizations of Pauli matrices DFT matrix Circulant matrix Weyl, H. (1927). "Quantenmechanik und Gruppentheorie". Zeitschrift
Generalized_Clifford_algebra
_{i}^{p^{m_{i}-2}}\\\end{bmatrix}},} which is a circulant matrix. It is well known that a circulant matrix-vector product can be efficiently computed by
Cyclotomic fast Fourier transform
Cyclotomic_fast_Fourier_transform
convolution or wrapped convolution. It results from multiplication of a skew circulant matrix, generated by vector a, with vector b. Circular convolution theorem
Negacyclic_convolution
Polynomial of the elements of a matrix
of Z. This is a consequence of Z being a circulant matrix, as well as the theorem: If A is a circulant matrix in the class Ω(n, k) then if k > 3, perm(A) > |det
Permanent_(mathematics)
Mathematical concept in algebra
diagonal matrix commutes with all other diagonal matrices. Circulant matrices commute. They form a commutative ring since the sum of two circulant matrices
Commuting_matrices
In mathematics, invariant of square matrices
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
Determinant
Generalization of the discrete Fourier transform
used in numerical analysis. A circulant matrix is a matrix where every column is a cyclic shift of the previous one. Circulant matrices can be diagonalized
Fourier transform on finite groups
Fourier_transform_on_finite_groups
Integral expressing the amount of overlap of one function as it is shifted over another
fractional integral and fractional derivative. Analog signal processing Circulant matrix Convolution for optical broad-beam responses in scattering media Convolution
Convolution
Assignment problem in combinatorial mathematics
In the case of the ménage problem, the matrix arising from this view of the problem is the circulant matrix in which all but two adjacent elements of
Ménage_problem
Linear error correcting code
lower-cost hardware—in particular, codes constructed such that the H matrix is a circulant matrix. Yet another way of constructing LDPC codes is to use finite
Low-density_parity-check_code
Families of matrices in mathematics, physics, and quantum information
prime p Hermitian matrix Bloch sphere Discrete Fourier transform Generalized Clifford algebra Weyl–Brauer matrices Circulant matrix Shift operator Quantum
Generalizations of Pauli matrices
Generalizations_of_Pauli_matrices
Mapping involving integration between function spaces
kernel. Bateman transform Convolution kernel Circular convolution Circulant matrix Differential equations Kernel method List of transforms List of operators
Integral_transform
Polynomial whose coefficients are all 1 or −1
zero. The circulant matrix whose first row is ( a 0 , … , a N − 1 ) {\displaystyle (a_{0},\ldots ,a_{N-1})} is then a circulant Hadamard matrix. Ryser's
Littlewood_polynomial
definition of a circulant matrix, and ΛH is a diagonal matrix whose diagonal elements correspond to the first column of the circulant channel matrix H. The receiver
Carrier_interferometry
Function in discrete mathematics
consequence of the circular convolution theorem is that the DFT matrix F diagonalizes any circulant matrix. A useful property of the DFT is that the inverse DFT
Discrete_Fourier_transform
Algorithm for generating pseudo-randomized numbers
linear complexity test implemented in the TestU01 suite; a Boolean circulant matrix initialized from consecutive bits of an LFSR will never have rank greater
Linear_congruential_generator
Discrete Fourier transform for prime sizes
DFT matrix becomes a circulant matrix. Multiplying a data sequence with a circulant matrix is equivalent to the cyclic convolution with the matrix's row
Rader's_FFT_algorithm
Representation theory of groups
of order n, the matrix form of an element of K[C] acting on K[C] by multiplication takes a distinctive form known as a circulant matrix, in which each
Regular_representation
matrix Tridiagonal matrix Block matrix Sparse matrix Hessenberg matrix Hessian matrix Vandermonde matrix Stochastic matrix Toeplitz matrix Circulant matrix
Outline_of_linear_algebra
Vector satisfying some of the criteria of an eigenvector
algebra, a generalized eigenvector of an n × n {\displaystyle n\times n} matrix A {\displaystyle A} is a vector which satisfies certain criteria which are
Generalized_eigenvector
Square matrix that is a generalization of the Hadamard matrix
[\mathbf {A} _{1}]_{p}} are pxp Jacket matrix, then [ A ] N {\displaystyle [A]_{N}} is a block circulant matrix if and only if A 0 A 1 r t + A 1 r t A
Jacket_matrix
Number with an integer power equal to 1
group. The roots of unity appear as entries of the eigenvectors of any circulant matrix; that is, matrices that are invariant under cyclic shifts, a fact that
Root_of_unity
shown in 2019 that relaxing the symmetry and circulant requirements nevertheless permits a Hadamard matrix of this block form to exist for that order.
Williamson_conjecture
orbit B 6 ( θ ) {\displaystyle B_{6}(\theta )} , including the circulant Hadamard matrix C 6 {\displaystyle C_{6}} , a two-parameter orbit including the
Complex_Hadamard_matrix
Mathematical model which is both linear and time-invariant
p. 1. Welcome! Crutchfield, p. 1. Exercises Phillips 2007, p. 508. Circulant matrix Frequency response Impulse response System analysis Green function
Linear_time-invariant_system
Jabotinsky matrix (sometimes called iteration matrix or power matrix) is an infinite matrix used to convert function composition into matrix multiplication
Jabotinsky_matrix
analysis: Sparse matrix Band matrix Bidiagonal matrix Tridiagonal matrix Pentadiagonal matrix Skyline matrix Circulant matrix Triangular matrix Diagonally dominant
List of numerical analysis topics
List_of_numerical_analysis_topics
Number line and triangular tiling's symmetry mathematical structure
&\vdots \\0&0&0&\cdots &2&-1\\-1&0&0&\cdots &-1&2\end{array}}\right]} (a circulant matrix) for n > 2 {\displaystyle n>2} . Like other Kac–Moody algebras, affine
Affine_symmetric_group
matrix H {\displaystyle {\mathit {H}}} . Thus from the above property, we see that the core of E {\displaystyle {\mathit {E}}} is a circulant matrix consisting
Coding theory approaches to nucleic acid design
Coding_theory_approaches_to_nucleic_acid_design
Gerhard J. (1999), "The Steiner tree problem in Kalmanson matrices and in circulant matrices", Journal of Combinatorial Optimization, 3 (1): 51–58, doi:10
Kalmanson combinatorial conditions
Kalmanson_combinatorial_conditions
Graph where each vertex has the same number of neighbors
that are regular but not strongly regular are the cycle graph and the circulant graph on 6 vertices. The complete graph Km is strongly regular for any
Regular_graph
parallelogram equals the sum of the diagonals of the original quadrilateral. Circulant matrix Midpoint-stretching polygon Varignon's theorem Gardner 2006, p. 36
Midpoint_polygon
modulo a prime p {\displaystyle p} . The transformation operator is a circulant matrix. The number theoretic transform is meaningful in the ring Z m {\displaystyle
Number theoretic Hilbert transform
Number_theoretic_Hilbert_transform
into odd cycles. Their proof relies on Cayley graphs, in particular, circulant graphs, and many of their decompositions come from the action of a permutation
Cycle decomposition (graph theory)
Cycle_decomposition_(graph_theory)
Cryptographic operation in the Rijndael encryption algorithm
vector of four numbers in Rijndael's Galois field by the following circulant MDS matrix: [ d 0 d 1 d 2 d 3 ] = [ 2 3 1 1 1 2 3 1 1 1 2 3 3 1 1 2 ] [ b 0
Rijndael_MixColumns
Probability theory concept
simulation of stationary Gaussian processes through circulant embedding of the covariance matrix.", SIAM Journal on Scientific Computing, 18 (4): 1088–1107
Fractional_Brownian_motion
Discrete Fourier transform algorithm
algorithms and polynomial multiplication, efficient matrix–vector multiplication for Toeplitz, circulant and other structured matrices, filtering algorithms
Fast_Fourier_transform
Software packages using DDA
K. White (2020). "Accelerating the discrete dipole approximation via circulant preconditioning". J. Quant. Spectrosc. Radiat. Transfer. 240 106689. Bibcode:2020JQSRT
Discrete dipole approximation codes
Discrete_dipole_approximation_codes
Special type of Boolean function
whether a real circulant (cyclic) Hadamard matrix of order greater than 4 can exist at all is the subject of Ryser's conjecture on circulant Hadamard matrices
Bent_function
Type of block code
n} . Such a code is known as an s {\displaystyle s} -QC code. A double circulant code is a quasi-cyclic code of even length with s = 2 {\displaystyle s=2}
Cyclic_code
Graph of numbers differing by a square
{q-1}{4}}.} When q is prime, the associated Paley graph is a Hamiltonian circulant graph. Paley graphs are quasi-random: the number of times each possible
Paley_graph
Graph of chess rook moves
chessboards whose width and height are relatively prime, the rook's graphs are circulant graphs. With one exception, the rook's graphs can be distinguished from
Rook's_graph
construct Hadamard matrices. Ryser's conjecture on circulant Hadamard matrices: no real circulant Hadamard matrix has order greater than 4. Barker sequence conjecture:
List of unsolved problems in mathematics
List_of_unsolved_problems_in_mathematics
Method for computing radiation
n_{z}} , the method embeds the block Toeplitz array into a larger block circulant array of size ( 2 n x − 1 ) × ( 2 n y − 1 ) × ( 2 n z − 1 ) {\displaystyle
Discrete_dipole_approximation
Graph defined from a mathematical group
generally, the Cayley graphs of finite cyclic groups are exactly the circulant graphs. The Cayley graph of the direct product of groups (with the cartesian
Cayley_graph
Graph metric of electrical resistance between nodes
Heping; Yang, Yujun (2007). "Resistance distance and Kirchhoff index in circulant graphs". Int. J. Quantum Chem. 107 (2): 330–339. Bibcode:2007IJQC..107
Resistance_distance
American mathematician
S2CID 122353256. C. M. Ablow; J. L. Brenner (1963). "Roots and Canonical Forms for Circulant Matrices". Transactions of the American Mathematical Society. 107 (2):
Joel_Lee_Brenner
Graph without triples of adjacent vertices
(1/3 − ε)n for any ε > 0. Andrásfai graph, a family of triangle-free circulant graphs with diameter two Henson graph, an infinite triangle-free graph
Triangle-free_graph
Geolocation system
des dispositifs de signalement électronique et lumineux des aéronefs circulant sans personne à bord "Infosecurity Blogs". Infosecurity Magazine. Retrieved
Wi-Fi_positioning_system
American mathematician (1923–2018)
at the age of 95. Ancient Loons: Stories David Pingree Told Me (2016) Circulant matrices Descartes' Dream: The World According to Mathematics by Philip
Philip_J._Davis
American mathematician
Kalman, Dan; White, James E. (November 2001). "Polynomial Equations and Circulant Matrices" (PDF). American Mathematical Monthly. 108 (9): 821–841. doi:10
Dan_Kalman
Polish mathematician
doi:10.1017/S0013091500011214. Minc, Henryk (1964). "Permanents of (0, 1)-Circulants". Canadian Mathematical Bulletin. 7 (2): 253–263. doi:10.4153/CMB-1964-023-3
Henryk_Minc
Canadian mathematician
1974 under Herbert Ryser at Caltech with thesis Rational G-Circulants Satisfying the Matrix Equation A 2 = d I + λ J {\displaystyle A^{2}=dI+\lambda J}
Clement_W._H._Lam
New Zealand mathematician (1928–2013)
2307/2684063. JSTOR 2684063. 1970s Searle, S.R. (1979). "On inverting circulant matrices". Linear Algebra and Its Applications. 25: 77–89. doi:10
Shayle_R._Searle
Neural network coding model
model, in which a redundant dictionary is modeled as a concatenation of circulant matrices. While the global sparsity constraint describes signal x ∈ R
Convolutional_sparse_coding
Unsolved problem in computational complexity theory
log space, a class contained in P) Interval graphs Permutation graphs Circulant graphs Bounded-parameter graphs Graphs of bounded treewidth Graphs of
Graph_isomorphism_problem
Type of stochastic process
think of the correlation function as a linear operator. Since it is a circulant operator (depends only on the difference between the two arguments), its
Stationary_process
Method of encoding digital data on multiple carrier frequencies
channel matrices in (1) are pseudo-circulant and can be diagonalized by the M {\displaystyle M} -point DFT/IDFT matrix with some diagonal phase shift matrices
Orthogonal frequency-division multiplexing
Orthogonal_frequency-division_multiplexing
Medical imaging procedure
"CT-perfusion imaging of the human brain: Advanced deconvolution analysis using circulant singular value decomposition". Computerized Medical Imaging and Graphics
CT_scan
Theory of stochastic processes
optimal only when the covariance structure has the specific symmetry (circulant or centrosymmetric, respectively) that matches their basis functions;
Kosambi–Karhunen–Loève theorem
Kosambi–Karhunen–Loève_theorem
Mathematical object
, 0 ) {\displaystyle \mathbf {F} =(-1,0,\ldots ,0)} corresponding to circulant matrices, because all the coordinates of [F∗u]v are bounded by 1, and
Ideal_lattice
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