# 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

$$\left[\begin{array}{ccccc}\hfill {M}_{1}\hfill & \hfill {M}_{2}\hfill & \hfill {M}_{3}\hfill & \hfill \mathrm{\dots}\hfill & \hfill {M}_{d}\hfill \\ \hfill {M}_{d}\hfill & \hfill {M}_{1}\hfill & \hfill {M}_{2}\hfill & \hfill \mathrm{\dots}\hfill & \hfill {M}_{d-1}\hfill \\ \hfill {M}_{d-1}\hfill & \hfill {M}_{d}\hfill & \hfill {M}_{1}\hfill & \hfill \mathrm{\dots}\hfill & \hfill {M}_{d-2}\hfill \\ \hfill \mathrm{\vdots}\hfill & \hfill \mathrm{\vdots}\hfill & \hfill \mathrm{\vdots}\hfill & \hfill \mathrm{\ddots}\hfill & \hfill \mathrm{\vdots}\hfill \\ \hfill {M}_{2}\hfill & \hfill {M}_{3}\hfill & \hfill {M}_{4}\hfill & \hfill \mathrm{\dots}\hfill & \hfill {M}_{1}\hfill \end{array}\right]$$ |

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 |
---|---|

Canonical name | CirculantMatrix |

Date of creation | 2013-03-22 13:53:38 |

Last modified on | 2013-03-22 13:53:38 |

Owner | bwebste (988) |

Last modified by | bwebste (988) |

Numerical id | 9 |

Author | bwebste (988) |

Entry type | Definition |

Classification | msc 15-01 |

Classification | msc 15A99 |