[Math] Determinant of circulant matrix

Question

Find the determinant of the following matrix in the terms of $a_1,a_2,\cdots,a_n$ explicitly,
$$
\begin{bmatrix}
a_1 & a_2 & a_3 & \cdots & a_n\\
a_2 & a_3 & a_4 & \cdots & a_1\\
a_3 & a_4 & a_5 & \cdots & a_2\\
\vdots & \vdots & \vdots & \ddots & \vdots\\
a_n & a_1 & a_2 & \cdots & a_{n-1}\\
\end{bmatrix}
$$
When the determinant is zero?

Answer 1

Call that matrix $A$ and notice that it is a permutation of a circulant matrix, $$ A = CP $$ Where $P$ is a permutation matrix with ones on the anti-diagonal, and zeros in all other positions. Then

$$ \det[A] = \det[CP] = \det[C]\det[P] $$ The determinant of the permutation part can be shown to depend on the size $n$. It can be written as $$ \det[P] = (-1)^{\left\lfloor\frac{n}{2}\right\rfloor} $$ Now $C$ is $$ \begin{bmatrix} a_{n} & a_{n-1} & a_{n-2} & \cdots & a_1\\ a_{1} & a_{n} & a_{n-1} & \cdots & a_2\\ a_{2} & a_{1} & a_n & \cdots & a_3\\ \vdots & \vdots & \vdots & \ddots & \vdots\\ a_{n-1} & a_{n-2} & a_{n-3} & \cdots & a_{n}\\ \end{bmatrix} $$ $C$ is a circulant matrix. Define the associated polynomial $$ f(\omega) = a_n + \sum_{k=1}^{n-1} a_k\omega^k $$ Then using the product formula on the Wikipedia page for circulant matrices, $$ \det[C] = \prod_{j=0}^{n-1}f(\omega_j), $$ where $\omega_j=e^{\frac{2\pi i j}{n}}$ and $i=\sqrt{-1}$. Then the final formula is $$ \det[A] = (-1)^{\left\lfloor\frac{n}{2}\right\rfloor}\prod_{j=0}^{n-1}\left(a_n + \sum_{k=1}^{n-1} a_k\omega_j^k\right) $$