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Mathematics-Online course: Linear Algebra - Matrices - Special Matrices

Cyclic Matrices

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In a cyclic $ n\times n$-matrix every column is a cyclic permutation of the first column as follows:

\begin{displaymath} \left[ \begin{array}{cccc} a_0 & a_{n-1} & a_{n-2} & \ldo... ...dots\\ \vdots & \vdots & \vdots & \end{array} \right]\,. \end{displaymath}

The product of two cyclic $ n\times n$-matrices is cyclic.
(Authors: App/Burkhardt/Höllig)
The entries of a product of cyclic matrices $ C = AB$ are

$\displaystyle c_{i,k} = \sum_{j=1}^n a_{i-j\operatorname{mod}n} b_{j-k\operatorname{mod}n}\, . $

Since

$\displaystyle \{0,\ldots,n-1\} = k+\{0,\ldots,n-1\} \operatorname{mod}n\, , $

we can replace $ j$ with $ j+k$ in the addends. This proves that $ c_{i,k}$ depends only on $ i-k\operatorname{mod}n$.
(Authors: App/Burkhardt/Höllig)
The product of teh cyclic matrices

\begin{displaymath} A=\left( \begin{array}{ccc} -3 & 4 & 1\\ 1 & -3 & 4\\ ... ... & -2 & 5\\ 5 & 0 & -2\\ -2 & 5 & 0 \end{array} \right) \end{displaymath}

is the cyclic matrix

\begin{displaymath} C=AB=\left( \begin{array}{ccc} 18 & 11 & -23\\ -23 & 18 & 11\\ 11 & -23 & 18 \end{array} \right)\,. \end{displaymath}

(Authors: App/Burkhardt/Höllig)
  automatically generated 4/21/2005