Tagged: root of unity

Determinant of a General Circulant Matrix

Problem 374

Let \[A=\begin{bmatrix}
a_0 & a_1 & \dots & a_{n-2} &a_{n-1} \\
a_{n-1} & a_0 & \dots & a_{n-3} & a_{n-2} \\
a_{n-2} & a_{n-1} & \dots & a_{n-4} & a_{n-3} \\
\vdots & \vdots & \dots & \vdots & \vdots \\
a_{2} & a_3 & \dots & a_{0} & a_{1}\\
a_{1} & a_2 & \dots & a_{n-1} & a_{0}
\end{bmatrix}\] be a complex $n \times n$ matrix.
Such a matrix is called circulant matrix.
Then prove that the determinant of the circulant matrix $A$ is given by
\[\det(A)=\prod_{k=0}^{n-1}(a_0+a_1\zeta^k+a_2 \zeta^{2k}+\cdots+a_{n-1}\zeta^{k(n-1)}),\] where $\zeta=e^{2 \pi i/n}$ is a primitive $n$-th root of unity.

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Group of $p$-Power Roots of 1 is Isomorphic to a Proper Quotient of Itself

Problem 221

Let $p$ be a prime number. Let
\[G=\{z\in \C \mid z^{p^n}=1\} \] be the group of $p$-power roots of $1$ in $\C$.

Show that the map $\Psi:G\to G$ mapping $z$ to $z^p$ is a surjective homomorphism.
Also deduce from this that $G$ is isomorphic to a proper quotient of $G$ itself.

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Finite Order Matrix and its Trace

Problem 28

Let $A$ be an $n\times n$ matrix and suppose that $A^r=I_n$ for some positive integer $r$. Then show that

(a) $|\tr(A)|\leq n$.

(b) If $|\tr(A)|=n$, then $A=\zeta I_n$ for an $r$-th root of unity $\zeta$.

(c) $\tr(A)=n$ if and only if $A=I_n$.

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