Posts Tagged ‘Circulant Matrices’

## Circulant Matrices

July 17, 2009
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**Definition.** Given be a linear operator such that for every then

The matrix with k-row is given by will be called circulant and denoted by

**Theorem 1.** , where is a primitive n-th root of unity.

We multiply the j-columm with , and add it onto the first columm, claim that , we have:

The j-element of the first columm is a product of with . We give as a polynomial with n variable on , hence is devided by . Claim that is a n-degree polynomial and the coefficient of is 1.

Thus

For , we denote and

We can easily calculate and receive that and since is a linearly independent system, we conclude that are eigenvalues of circulant matrix

Some results:

i.

ii. With non-singular matrix

then

iii where

(To be continued)

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Categories: Matrix Theory
Circulant Matrices, Matrix Theory

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