Difference between revisions of "Circulant matrix"
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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}} | + | * 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 | + | * 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
Circulant matrix. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Circulant_matrix&oldid=37546