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Difference between revisions of "Circulant matrix"

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(Start article: Circulant matrix)
 
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==References==
 
==References==
* Marcus, Marvin, Minc, Henryk ''A survey of matrix theory and matrix inequalities'' Dover (1969)[1964] ISBN 0-486-67102-X {{ZBL|0126.02404}}
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* Marcus, Marvin, Minc, Henryk ''A survey of matrix theory and matrix inequalities'' Dover (1969)[1964] {{ISBN|0-486-67102-X}} {{ZBL|0126.02404}}
* Muir, Thomas ''A treatise on the theory of determinants''. Dover Publications (1960) [1933] ISBN 0-486-60670-8
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* Muir, Thomas ''A treatise on the theory of determinants''. Dover Publications (1960) [1933] {{ISBN|0-486-60670-8}}

Latest revision as of 19:40, 17 November 2023

A square matrix in which the rows are successive cyclic shifts of the first. The term circulant may denote such a matrix or the determinant of such a matrix.

Let denote the n \times n circulant matrix with entries C_{12} = C_{23} = \cdots = C_{n-1,n} = C_{n1} = 1 and all other entries zero. If \zeta is an n-th root of unity then the vector v_\zeta = (1,\zeta,\ldots,\zeta^{n-1})^\top is an eigenvector of C with eigenvalue \zeta. Further, a general circulant with first row (a_0, a_1, \ldots, a_{n-1}) is equal to the polynomial a(C) = a_0 I + a_1 C + \cdots + a_{n-1} C^{n-1}. Hence all circulant matrices commute, and have v_\zeta as a common eigenvector with corresponding eigenvalue a(\zeta).

References

  • Marcus, Marvin, Minc, Henryk A survey of matrix theory and matrix inequalities Dover (1969)[1964] ISBN 0-486-67102-X Zbl 0126.02404
  • Muir, Thomas A treatise on the theory of determinants. Dover Publications (1960) [1933] ISBN 0-486-60670-8
How to Cite This Entry:
Circulant matrix. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Circulant_matrix&oldid=37546