Circulant Matrices
Why
1Definition
Define $S \in \R ^{d \times d}$ by
\[ S = \bmat{ 0 & 0 & \dots & 0 & 1 \\ 0 & 1 & \cdots & 0 & 0 \\ 0 & 0 & \ddots & 0 & 0 \\ 0 & 0 & \cdots & 1 & 0 \\ } \]
For a vector $x \in \R ^d$ the down shift of $x$ is $Sx$.Let $A \in \R ^{d \times d}$ be a matrix with columsn $a_1, \dots , a_d$. $A$ is a circulant matrix if $a_1 = Sa_d$, $a2 = Sa_2$, and $a_i = A_{i -1}$ for $i = 2, \dots , d$.
Example
For example, the matrix
\[
\bmat{
1 & 4 & 3 & 2 \\
2 & 1 & 4 & 3 \\
3 & 2 & 1 & 4 \\
4 & 3 & 2 & 1 \\
}
\]
Characterization
A matrix $C \in \R ^{d \times d}$ is circulant
if and only if there exists $c_0, \dots ,
c_{d-1}$ so that
\[
C = c_0I + c_1 S + c_2S^2 + \cdot s + c_{n-1}S^{n-1}.
\]
Properties
The sum and product of any two circulant matrix is circulant. In other words, the circulant matrices with the usual matrix addition and multiplication form a commutative ring.
- Future sheets will include. These matrices arise in practice and each has the same eigenvectors. ↩︎