Read More
Date: 28-9-2021
![]()
Date: 1-12-2021
![]()
Date: 30-11-2021
![]() |
An matrix whose rows are composed of cyclically shifted versions of a length-
list
. For example, the
circulant matrix on the list
is given by
![]() |
(1) |
Circulant matrices are very useful in digital image processing, and the circulant matrix is implemented as CirculantMatrix[l, n] in the Mathematica application package Digital Image Processing.
Circulant matrices can be implemented in the Wolfram Language as follows.
CirculantMatrix[l_List?VectorQ] :=
NestList[RotateRight, RotateRight[l],
Length[l] - 1]
CirculantMatrix[l_List?VectorQ, n_Integer] :=
NestList[RotateRight,
RotateRight[Join[Table[0, {n - Length[l]}],
l]], n - 1] /; n >= Length[l]
where the first input creates a matrix with dimensions equal to the length of and the second pads with zeros to give an
matrix. A special type of circulant matrix is defined as
![]() |
(2) |
where is a binomial coefficient. The determinant of
is given by the beautiful formula
![]() |
(3) |
where ,
, ...,
are the
th roots of unity. The determinants for
, 2, ..., are given by 1,
, 28,
, 3751, 0, 6835648,
, 364668913756, ... (OEIS A048954), which is 0 when
.
Circulant matrices are examples of Latin squares.
REFERENCES:
Davis, P. J. Circulant Matrices, 2nd ed. New York: Chelsea, 1994.
Sloane, N. J. A. Sequences A048954 and A049287 in "The On-Line Encyclopedia of Integer Sequences."
Stroeker, R. J. "Brocard Points, Circulant Matrices, and Descartes' Folium." Math. Mag. 61, 172-187, 1988.
Vardi, I. Computational Recreations in Mathematica. Reading, MA: Addison-Wesley, p. 114, 1991.
|
|
اكتشاف الخرف مبكرا بعلامتين.. تظهران قبل 11 عاما
|
|
|
|
|
العراق يفوز بمنصب نائب رئيس لجنة الشرق الأوسط لمنظمة الأمم المتحدة للسياحة
|
|
|
|
|
قسم بين الحرمين يعلن عن جاهزيته لاستقبال زائري النصف من شعبان
|
|
|