Linear and multilinear algebra; matrix theory Example

Circulant matrix

Example

Let $F$ be a field and let $c_0, c_1, \dots, c_{n-1} \in F$ . The circulant matrix with first column $(c_0, c_1, \dots, c_{n-1})^T$ is the $n \times n$ matrix $C$ with entries $C_{ij} = c_{(i - j) \bmod n}$ , for $0 \le i, j \le n - 1$ . Each column is the previous column shifted down by one place, with the last entry moved to the top. For $n = 3$ : $C = \begin{pmatrix} c_0 & c_2 & c_1 \\ c_1 & c_0 & c_2 \\ c_2 & c_1 & c_0 \end{pmatrix}.$

Remarks

Some authors define circulant matrices by their first row; the two conventions give transposed matrices. Over $\C$ , the vectors $(1, \omega^k, \omega^{2k}, \dots, \omega^{(n-1)k})$ with $\omega = e^{2\pi i/n}$ are eigenvectors of every $n \times n$ circulant matrix. The product of two circulant matrices of the same order is circulant, and circulant matrices commute with each other.

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Matrix definition
Square matrix definition
Circulant matrix example

Tag	`07HFRH2AWY`
URI	`m2://07HFRH2AWY`
Kind	example
Writing	W2 usable
MSC	15B05
Instance of	Square matrix
Uses	Square matrix
Same as	Wikidata Q245457 Wikipedia Circulant matrix

Circulant matrix

Example

Remarks

Reading path

Further reading

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