# circulant matrix

A square matrix $M:A\times A\to C$ is said to be $g$-circulant for an integer $g$ if each row other than the first is obtained from the preceding row by shifting the elements cyclically g columns to the right (g¿0) or -g columns to the left (g ¡ 0).

That is, if $A=[a_{ij}]$ then $a_{i,j}=a_{i+1,j+g}$ where the subscripts are computed modulo d. A 1-circulant is commonly called a circulant and a -1-circulant is called a back circulant.

More explicitly, a matrix of the form

 $\left[\begin{array}[]{ccccc}M_{1}&M_{2}&M_{3}&\ldots&M_{d}\\ M_{d}&M_{1}&M_{2}&\ldots&M_{d-1}\\ M_{d-1}&M_{d}&M_{1}&\ldots&M_{d-2}\\ \vdots&\vdots&\vdots&\ddots&\vdots\\ M_{2}&M_{3}&M_{4}&\ldots&M_{1}\end{array}\right]$

is called circulant.

Because the Jordan decomposition (http://planetmath.org/JordanCanonicalFormTheorem) of a circulant matrix is rather simple, circulant matrices have some interest in connection with the approximation of eigenvalues of more general matrices. In particular, they have become part of the standard apparatus in the computerized analysis of signals and images.

Title circulant matrix CirculantMatrix 2013-03-22 13:53:38 2013-03-22 13:53:38 bwebste (988) bwebste (988) 9 bwebste (988) Definition msc 15-01 msc 15A99