Matrices of Tangents and Cotangents and Their Associated Integer Matrices

Abstract

In this paper, we consider certain special sine, tangent, and cotangent symmetric matrices, along with their associated integer matrices. We present interesting relationships that connect these matrices. Relevant examples of the corresponding matrices are also provided, along with several intriguing problems. A separate section includes proofs of the main theorems, which we believe are noteworthy in their own right.

1. Introduction and Main Results

The starting point for the matrices considered in this work, which are exclusively square matrices, was the matrices of sines and cosines, along with the easy-to-determine identities for the determinants of these matrices, presented in Appendix A. The matrices of tangents and cotangents are more difficult to discuss. While seeking analogous formulas for determinants of these matrices, we discovered two families of integer matrices and nontrivial relationships between them and certain matrices of sines. The determinants of these integer matrices belong to the set of modular determinants, which have attracted the attention of mathematicians for a long time [1,2,3,4,5,6]. Modular determinants have subsequently found applications in elementary number theory, computational number theory, cryptography, and bioinformatics. Based on the theory of trigonometric Fourier series, the aforementioned matrices of sines were used to evaluate the sum of some alternating series [7,8,9,10]. Moreover, the matrices studied here provide interesting examples of bounded operators with noteworthy spectra; this topic will be investigated in a separate paper.

Let $\mathbb{P}$ be the set of prime numbers, and let $p\in \mathbb{P}$, $p\ge 3$. For any number $z\in \mathbb{Z}$, let ${mod}_{p}z\in \{0,1,\dots ,p-1\}$ represent the remainder when dividing z by p, which by definition satisfies $p\mid (z-{mod}_{p}z)$. For $z\in \{1,2,\dots ,p-1\}$, let ${inv}_{p}z\in \{1,2,\dots ,p-1\}$ denote the inverse of z modulo p, such that ${mod}_p\left(z{inv}_{p}z\right)=1$. Note that because p is prime, the inverse modulo p always exists and is uniquely determined.

First, we will define three families of matrices, namely the matrices of tangents, cotangents, and sines, which are the focus of this paper.

Definition 1. 

Let $p\in \mathbb{P}$, $p\ge 3$. We define the following matrices from ${\mathcal{M}}_{{\displaystyle \frac{p-1}{2}}}\left(\mathbb{R}\right)$

$$T_p:={\left[tan{\displaystyle \frac{ij\pi }{p}}\right]}_{{\displaystyle \frac{p-1}{2}}\times {\displaystyle \frac{p-1}{2}}},C_p:={\left[cot{\displaystyle \frac{ij\pi }{p}}\right]}_{{\displaystyle \frac{p-1}{2}}\times {\displaystyle \frac{p-1}{2}}},$$

and

$$S_p:={\left[sin{\displaystyle \frac{2ij\pi }{p}}\right]}_{{\displaystyle \frac{p-1}{2}}\times {\displaystyle \frac{p-1}{2}}}.$$

Clearly, the matrices $T_p$, $C_p$ and $S_p$ are symmetric. Using these matrices, we define two additional matrices that possess very interesting properties.

Definition 2. 

Let $p\in \mathbb{P}$, $p\ge 3$. The matrices ${\mathsf{\Phi }}_p,{\mathsf{\Psi }}_p\in {\mathcal{M}}_{{\displaystyle \frac{p-1}{2}}}\left(\mathbb{R}\right)$ are defined by the products

$${\mathsf{\Phi }}_p:={\displaystyle \frac{2}{p}}{S}_p{T}_p\mathrm{and}{\mathsf{\Psi }}_p:=2S_p{C}_p.$$

(1)

The matrices given by formulas (1) will be referred to as matrices associated with the matrices $T_p$ and $C_p$, respectively.

As will be shown later, ${\mathsf{\Phi }}_p$ contains only 1s, up to a sign, while ${\mathsf{\Psi }}_p$ contains, up to a sign, a sequence of numbers $1,3,\dots ,p-2$ in some order, in every row and every column.

Remark 1. 

Note that the restriction of our discussion to $p\in \mathbb{P}$, $p\ge 3$, is not arbitrary; it is a natural assumption. For example, the matrices $T_8$ and $T_{12}$ do not exist, while the matrices $T_6$, $T_9$, and $T_{10}$ do exist, but

$${\mathsf{\Phi }}_6=\left[\begin{array}{cc}{\displaystyle \frac{2}{3}}& 0\\ -{\displaystyle \frac{1}{3}}& 1\end{array}\right],{\mathsf{\Phi }}_9=\left[\begin{array}{cccc}1& 1& 0& 1\\ -1& 1& 0& 1\\ 1& -1& 1& 1\\ -1& -1& 0& 1\end{array}\right],{\mathsf{\Phi }}_{10}=\left[\begin{array}{cccc}{\displaystyle \frac{4}{5}}& 0& -{\displaystyle \frac{2}{5}}& 0\\ -{\displaystyle \frac{3}{5}}& 1& -{\displaystyle \frac{1}{5}}& 1\\ {\displaystyle \frac{2}{5}}& 0& {\displaystyle \frac{4}{5}}& 0\\ -{\displaystyle \frac{1}{5}}& -1& {\displaystyle \frac{3}{5}}& 1\end{array}\right].$$

Similarly, the matrices $C_9$ and $C_{12}$ do not exist, while the matrices $C_6$, $C_8$, and $C_{10}$ do exist, but

$${\mathsf{\Psi }}_6=\left[\begin{array}{cc}4& 0\\ 2& 2\end{array}\right],{\mathsf{\Psi }}_8=\left[\begin{array}{ccc}6& 0& 2\\ 4& 4& -4\\ 2& 0& 6\end{array}\right],{\mathsf{\Psi }}_{10}=\left[\begin{array}{cccc}8& 0& -4& 0\\ 6& 6& 2& -2\\ 4& 0& 8& 0\\ 2& 2& -6& 6\end{array}\right].$$

In the definitions of $T_p$, $C_p$, $S_p$, ${\mathsf{\Phi }}_p$, and ${\mathsf{\Psi }}_p$ for $p\in \mathbb{N}\setminus \mathbb{P}$, $p\ge 4$, we replace ${\displaystyle \frac{p-1}{2}}$ with its integer part $⌊{\displaystyle \frac{p-1}{2}}⌋$. Moreover, it is easy to observe that the matrices $T_{4n}$, where $n\in \mathbb{N}$, $n\ge 2$, do not exist, as nor do the matrices $C_{3n}$, where $n\in \mathbb{N}$, $n\ge 3$. In the first case, this results from the form of the entry in the second row and the n-th column, and in the second case from the form of the entry in the third row and the n-th column.

Before we proceed to the theorems concerning the matrices ${\mathsf{\Phi }}_p$ and ${\mathsf{\Psi }}_p$, we will define certain mappings based on the concept of remainders modulo p and present a surprising property of one of these mappings.

Definition 3. 

Let $p\in \mathbb{P}$, $p\ge 3$. The mappings

$$\begin{array}{cc}\hfill & {\sigma }_p:{\left\{1,2,\dots ,{\displaystyle \frac{p-1}{2}}\right\}}^2\to \{1,2,\dots ,p-1\},\hfill \\ \hfill & {\phi }_p:{\left\{1,2,\dots ,{\displaystyle \frac{p-1}{2}}\right\}}^2\to \{\pm 1\},\hfill \\ \hfill & {\psi }_p:{\left\{1,2,\dots ,{\displaystyle \frac{p-1}{2}}\right\}}^2\to \{\pm 1,\pm 3,\dots ,\pm (p-2)\}\hfill \end{array}$$

are defined as follows

$$\begin{array}{cc}\hfill {\sigma }_p(i,j)& ={mod}_p\left(2i{inv}_{p}j\right),\hfill \\ \hfill {\phi }_p(i,j)& =\left\{\begin{array}{cc}-1,\hfill & \mathit{if}{\displaystyle \frac{{\sigma }_p(i,j)}{2}}\mathit{or}{\displaystyle \frac{{\sigma }_p(i,j)+p}{2}}\mathit{is}\mathit{even},\hfill \\ 1,\hfill & \mathit{if}{\displaystyle \frac{{\sigma }_p(i,j)}{2}}\mathit{or}{\displaystyle \frac{{\sigma }_p(i,j)+p}{2}}\mathit{is}\mathit{odd},\hfill \end{array}\right.\hfill \\ \hfill {\psi }_p(i,j)& =\left\{\begin{array}{cc}p-{\sigma }_p(i,j),\hfill & \mathit{if}{\sigma }_p(i,j)\mathit{is}\mathit{even},\hfill \\ -{\sigma }_p(i,j),\hfill & \mathit{if}{\sigma }_p(i,j)\mathit{is}\mathit{odd}.\hfill \end{array}\right.\hfill \end{array}$$

Remark 2. 

In this article, by ${\left\{1,2,\dots ,{\displaystyle \frac{p-1}{2}}\right\}}^2$ we understand the Cartesian product

$$\left\{1,2,\dots ,{\displaystyle \frac{p-1}{2}}\right\}\times \left\{1,2,\dots ,{\displaystyle \frac{p-1}{2}}\right\}.$$

Remark 3. 

The values of the mappings ${\sigma }_p$, ${\phi }_p$, and ${\psi }_p$, where $p\in \{5,7,11\}$, are shown in the tables below (the rows correspond to the first argument and the columns to the second argument).

$$\begin{array}{cccccc}\hfill & \begin{array}{ccc}{\sigma }_5& 1& 2\\ 1& 2& 4\\ 2& 1& 2\end{array}\hfill & \hfill & \begin{array}{cccc}{\sigma }_7& 1& 2& 3\\ 1& 2& 1& 3\\ 2& 4& 2& 6\\ 3& 6& 3& 2\end{array}\hfill & \hfill & \begin{array}{cccccc}{\sigma }_{11}& 1& 2& 3& 4& 5\\ 1& 2& 1& 8& 6& 7\\ 2& 4& 2& 5& 1& 3\\ 3& 6& 3& 2& 7& 10\\ 4& 8& 4& 10& 2& 6\\ 5& 10& 5& 7& 8& 2\end{array}\hfill \\ \hfill & \begin{array}{ccc}{\phi }_5& 1& 2\\ 1& 1& 1\\ 2& -1& 1\end{array}\hfill & \hfill & \begin{array}{cccc}{\phi }_7& 1& 2& 3\\ 1& 1& -1& 1\\ 2& -1& 1& 1\\ 3& 1& 1& 1\end{array}\hfill & \hfill & \begin{array}{cccccc}{\phi }_{11}& 1& 2& 3& 4& 5\\ 1& 1& -1& -1& 1& 1\\ 2& -1& 1& -1& -1& 1\\ 3& 1& 1& 1& 1& 1\\ 4& -1& -1& 1& 1& 1\\ 5& 1& -1& 1& -1& 1\end{array}\hfill \\ \hfill & \begin{array}{ccc}{\psi }_5& 1& 2\\ 1& 3& -1\\ 2& 1& 3\end{array}\hfill & \hfill & \begin{array}{cccc}{\psi }_7& 1& 2& 3\\ 1& 5& -1& -3\\ 2& 3& 5& 1\\ 3& 1& -3& 5\end{array}\hfill & \hfill & \begin{array}{cccccc}{\psi }_{11}& 1& 2& 3& 4& 5\\ 1& 9& -1& 3& 5& -7\\ 2& 7& 9& -5& -1& -3\\ 3& 5& -3& 9& -7& 1\\ 4& 3& 7& 1& 9& 5\\ 5& 1& -5& -7& 3& 9\end{array}\hfill \end{array}$$

The symbols for the mappings ${\phi }_p$ and ${\psi }_p$ are chosen to be consistent with the matrices ${\mathsf{\Phi }}_p$ and ${\mathsf{\Psi }}_p$, respectively. Their connection with these matrices will be shown in Theorem 2.

Theorem 1. 

Let $p\in \mathbb{P}$, $p\ge 3$. For any $i,j\in \left\{1,2,\dots ,{\displaystyle \frac{p-1}{2}}\right\}$, each of the mappings $|{\psi }_p(i,·)|$ and $|{\psi }_p(·,j)|$ is a bijection. Specifically, for a fixed $i\in \left\{1,2,\dots ,{\displaystyle \frac{p-1}{2}}\right\}$, the mapping

$$\left\{1,2,\dots ,{\displaystyle \frac{p-1}{2}}\right\}\ni j\mapsto \left|{\psi }_p(i,j)\right|\in \{1,3,\dots ,p-2\}$$

is a bijection. Similarly, for a fixed $j\in \left\{1,2,\dots ,{\displaystyle \frac{p-1}{2}}\right\}$, the mapping

$$\left\{1,2,\dots ,{\displaystyle \frac{p-1}{2}}\right\}\ni i\mapsto \left|{\psi }_p(i,j)\right|\in \{1,3,\dots ,p-2\}$$

is also a bijection.

Now, we can proceed to the main theorems of this paper.

Theorem 2. 

Let $p\in \mathbb{P}$, $p\ge 3$. Then, ${\mathsf{\Phi }}_p,{\mathsf{\Psi }}_p\in {\mathcal{M}}_{{\displaystyle \frac{p-1}{2}}}\left(\mathbb{Z}\right)$ and the following equalities hold:

$${\mathsf{\Phi }}_p={\left[{\phi }_p(i,j)\right]}_{{\displaystyle \frac{p-1}{2}}\times {\displaystyle \frac{p-1}{2}}}\mathit{and}{\mathsf{\Psi }}_p={\left[{\psi }_p(i,j)\right]}_{{\displaystyle \frac{p-1}{2}}\times {\displaystyle \frac{p-1}{2}}}.$$

Proposition 1. 

As mentioned earlier, based on the definitions of the mappings ${\phi }_p$ and ${\psi }_p$, and relying on Theorem 1 in the second case, the matrix ${\mathsf{\Phi }}_p$, up to a sign, contains only 1s, while the matrix ${\mathsf{\Psi }}_p$, also up to a sign, includes an injective sequence of numbers $1,3,\dots ,p-2$ in every row and every column. Furthermore, for every $p\in \mathbb{P}$, $p\ge 3$, the diagonals of ${\mathsf{\Phi }}_p$ and ${\mathsf{\Psi }}_p$ form the vectors $[1,1,\dots ,1]$ and $[p-2,p-2,\dots ,p-2]$, respectively. Additionally, the last column of ${\mathsf{\Phi }}_p$ and the first column of ${\mathsf{\Psi }}_p$ contain the vectors $[1,1,\dots ,1]$ and $[p-2,p-4,\dots ,3,1]$, respectively.

Remark 4. 

Consider the explicit forms of the associated matrices ${\mathsf{\Phi }}_5$, ${\mathsf{\Phi }}_7$, ${\mathsf{\Phi }}_{11}$, ${\mathsf{\Psi }}_5$, ${\mathsf{\Psi }}_7$, and ${\mathsf{\Psi }}_{11}$:

$$\begin{array}{cc}\hfill {\mathsf{\Phi }}_5& =\left[\begin{array}{cc}1& 1\\ -1& 1\end{array}\right],{\mathsf{\Phi }}_7=\left[\begin{array}{ccc}1& -1& 1\\ -1& 1& 1\\ 1& 1& 1\end{array}\right],{\mathsf{\Phi }}_{11}=\left[\begin{array}{ccccc}1& -1& -1& 1& 1\\ -1& 1& -1& -1& 1\\ 1& 1& 1& 1& 1\\ -1& -1& 1& 1& 1\\ 1& -1& 1& -1& 1\end{array}\right],\hfill \\ \hfill {\mathsf{\Psi }}_5& =\left[\begin{array}{cc}3& -1\\ 1& 3\end{array}\right],{\mathsf{\Psi }}_7=\left[\begin{array}{ccc}5& -1& -3\\ 3& 5& 1\\ 1& -3& 5\end{array}\right],{\mathsf{\Psi }}_{11}=\left[\begin{array}{ccccc}9& -1& 3& 5& -7\\ 7& 9& -5& -1& -3\\ 5& -3& 9& -7& 1\\ 3& 7& 1& 9& 5\\ 1& -5& -7& 3& 9\end{array}\right].\hfill \end{array}$$

Theorem 3. 

If $p\in \mathbb{P}$, $p\ge 3$, then the following equalities holds:

$$T_p=2S_p{\mathsf{\Phi }}_p\mathit{and}{C}_p={\displaystyle \frac{2}{p}}{S}_p{\mathsf{\Psi }}_p.$$

Theorem 4. 

If $p\in \mathbb{P}$, $p\ge 3$, then

$$det{T}_p={(-1)}^{{\displaystyle \frac{(p-1)(p-3)}{8}}}{p}^{{\displaystyle \frac{p-1}{4}}}det{\mathsf{\Phi }}_p\mathit{and}det{C}_p={(-1)}^{{\displaystyle \frac{(p-1)(p-3)}{8}}}{p}^{{\displaystyle \frac{1-p}{4}}}det{\mathsf{\Psi }}_p.$$

Remark 5. 

The coefficient ${(-1)}^{{\displaystyle \frac{(p-1)(p-3)}{8}}}$ in Theorem 4 (and also in Lemma 1) is related to the ordering of the rows (or columns) of these matrices. If the rows (or columns) were arranged in reverse order, this coefficient would disappear.

Remark 6. 

Let us note that Theorems 2 and 4 lead to several interesting facts (for any $p\in \mathbb{P}$, $p\ge 3$):

(a) 

$det{T}_p\in \mathbb{Z}\iff p\in 4\mathbb{N}-3$;

(b) 

$\sqrt{p}det{T}_p\in \mathbb{Z}\iff p\in 4\mathbb{N}-1$;

(c) 

$p^{{\displaystyle \frac{p-1}{4}}}det{C}_p\in \mathbb{Z}$;

(d) 

$det{T}_{p}det{C}_p=det{\mathsf{\Phi }}_{p}det{\mathsf{\Psi }}_p$;

(e) 

the entries of $T_p$ and $C_p$ are irrational numbers, and $det{T}_p$ depends on $det{\mathsf{\Phi }}_p$, while $det{C}_p$ depends on $det{\mathsf{\Psi }}_p$, with the entries of ${\mathsf{\Phi }}_p$ and ${\mathsf{\Psi }}_p$ being integers.

Theorem 5. 

For every $p\in \mathbb{P}$, $p\ge 3$, the following identities hold:

$$T_p{C}_p={\mathsf{\Phi }}_p^{*}{\mathsf{\Psi }}_p\mathit{and}{C}_p{T}_p={\mathsf{\Psi }}_p^{*}{\mathsf{\Phi }}_p.$$

Remark 7. 

Note that the equalities from Theorem 5 are equivalent, since conjugating one equation yields the other.

Remark 8. 

Theorems 3–5 remain valid also for those composite integers $p\ge 3$ for which the matrices $T_p$ and $C_p$ exist, when $T_p$, $C_p$, and $S_p$ are defined as in Remark 1. Theorems 1 and 2 hold only for prime $p\ge 3$, which follows from Definition 3 and the function ${inv}_p$ used there.

Remark 9. 

In the article [11], we find the matrices ${\tilde{T}}_n,{\tilde{C}}_n\in {\mathcal{M}}_{|{\mathcal{I}}_n|}\left(\mathbb{R}\right)$, $n\in \mathbb{N}$, $n\ge 3$, defined as follows:

$${\tilde{T}}_n:={\left[tan{\displaystyle \frac{ij\pi }{n}}\right]}_{(i,j)\in {\mathcal{I}}_n^2},{\tilde{C}}_n:={\left[cot{\displaystyle \frac{ij\pi }{n}}\right]}_{(i,j)\in {\mathcal{I}}_n^2},$$

where

$${\mathcal{I}}_n=\left\{m\in \mathbb{N}:m\in \left\{1,2,\dots ,⌊{\displaystyle \frac{n-1}{2}}⌋\right\},gcd(m,n)=1\right\}.$$

The matrices ${\tilde{T}}_n$ and ${\tilde{C}}_n$ coincide with the matrices $T_n$ and $C_n$, respectively, when n is prime. The formulas for the determinant of ${\tilde{T}}_n$ and ${\tilde{C}}_n$ presented in the mentioned article are given from a number-theoretic point of view, as they are described using characters and the Euler totient function φ. More precisely, it is proved that

$$\begin{array}{cc}\hfill det{\tilde{T}}_n& =\pm {\left({\displaystyle \frac{n}{\pi }}\right)}^{{\displaystyle \frac{\phi \left(n\right)}{2}}}\prod _{\chi \in X_n^{-}}\sum _{n=1}^{\infty }{\displaystyle \frac{\chi \left(n\right)}{n}},\hfill \\ \hfill det{\tilde{C}}_n& =\pm {\left({\displaystyle \frac{n}{\pi }}\right)}^{{\displaystyle \frac{\phi \left(n\right)}{2}}}{N}_n\prod _{\chi \in X_n^{-}}\sum _{n=1}^{\infty }{\displaystyle \frac{\chi \left(n\right)}{n}},\hfill \end{array}$$

where $X_n^{-}$ is the set of odd Dirichlet characters modulo n, and if $n=2^{\alpha }k$, $gcd(2,k)=1$, $\alpha \in \mathbb{N}\cup \left\{0\right\}$, then

$$\begin{array}{cc}\hfill M_n& :=min\{r\in \mathbb{N}:2^r\equiv 1(modk)\},L_n:={\displaystyle \frac{\phi \left(k\right)}{M_n}},\hfill \\ \hfill N_n& :=\left\{\begin{array}{cc}1,\hfill & \mathit{if}n\equiv 0(mod4),\hfill \\ {(2^{M_n/2}+1)}^{L_n},\hfill & \mathit{if}n\equiv 1(mod2)\mathit{and}{M}_n\equiv 0(mod2),\hfill \\ {(2^{M_n}-1)}^{L_n/2},\hfill & \mathit{if}n\equiv 1(mod2)\mathit{and}{M}_n\equiv 1(mod2),\hfill \\ {(2^{M_n/2}+1)}^{-L_n},\hfill & \mathit{if}n\equiv 2(mod4)\mathit{and}{M}_n\equiv 0(mod2),\hfill \\ {(2^{M_n}-1)}^{-L_n/2},\hfill & \mathit{if}n\equiv 2(mod4)\mathit{and}{M}_n\equiv 1(mod2).\hfill \end{array}\right.\hfill \end{array}$$

2. Open Problems

We will now extend the results by presenting two problems.

Problem 1. 

Consider the following matrices:

$${\mathsf{\Phi }}_5^{*}{\mathsf{\Psi }}_5=\left[\begin{array}{cc}2& -4\\ 4& 2\end{array}\right],{\mathsf{\Phi }}_7^{*}{\mathsf{\Psi }}_7=\left[\begin{array}{ccc}3& -9& 1\\ -1& 3& 9\\ 9& 1& 3\end{array}\right],$$

and

$${\mathsf{\Phi }}_{11}^{*}{\mathsf{\Psi }}_{11}=\left[\begin{array}{ccccc}5& -25& 9& -7& 1\\ -1& 5& 7& -25& -9\\ -7& -9& 5& 1& 25\\ 9& -1& 25& 5& -7\\ 25& 7& 1& 9& 5\end{array}\right].$$

This raises the question of whether there is a simple way to describe the matrix ${\mathsf{\Phi }}_p^{*}{\mathsf{\Psi }}_p$. It can be hypothesized that for every $p\in \mathbb{P}$, $p\ge 3$, there exists an odd bijection between the corresponding entries of ${\mathsf{\Psi }}_p$ and ${\mathsf{\Phi }}_p^{*}{\mathsf{\Psi }}_p$.

Problem 2. 

Let us examine the following matrices:

$${\mathsf{\Phi }}_9=\left[\begin{array}{cccc}1& 1& 0& 1\\ -1& 1& 0& 1\\ 1& -1& 1& 1\\ -1& -1& 0& 1\end{array}\right]$$

and

$${\mathsf{\Phi }}_{25}=\left[\begin{array}{cccccccccccc}1& 1& 1& 1& 0& 1& -1& -1& -1& 0& -1& 1\\ -1& 1& 1& 1& 0& 1& 1& 1& 1& 0& 1& 1\\ 1& -1& 1& 1& 0& 1& -1& -1& 1& 0& 1& 1\\ -1& -1& -1& 1& 0& 1& -1& 1& -1& 0& -1& 1\\ 1& 1& -1& -1& 1& 1& 1& -1& -1& 1& 1& 1\\ -1& 1& -1& -1& 0& 1& -1& 1& 1& 0& 1& 1\\ 1& -1& 1& -1& 0& -1& 1& -1& 1& 0& -1& 1\\ -1& -1& 1& -1& 0& -1& 1& 1& -1& 0& 1& 1\\ 1& 1& 1& 1& 0& -1& -1& 1& 1& 0& 1& 1\\ -1& 1& -1& 1& -1& -1& 1& -1& 1& 1& -1& 1\\ 1& -1& -1& 1& 0& -1& 1& 1& -1& 0& 1& 1\\ -1& -1& -1& 1& 0& -1& -1& -1& -1& 0& 1& 1\end{array}\right].$$

The entries of the matrices ${\mathsf{\Phi }}_{49}$, ${\mathsf{\Phi }}_{121}$, and ${\mathsf{\Phi }}_{169}$ (the last of which is indeed very large) are also $-1$, 0, and 1, as verified using Wolfram Mathematica software. This naturally raises the question of whether, for every $p\in \mathbb{P}$, $p\ge 3$, the entries of ${\mathsf{\Phi }}_{p^2}$ consist only of the values $-1$, 0, and 1. A second, more general question concerns whether there is a simple description of the matrix ${\mathsf{\Phi }}_{p^2}$.

3. Lemmas and Proofs of Theorems

Proof of Theorem 1. 

To establish the validity of this theorem, it suffices to prove the injectivity of the mentioned mappings because the domain and codomain have the same finite cardinality.

First, fix $j\in \left\{1,2,\dots ,{\displaystyle \frac{p-1}{2}}\right\}$ and assume that for certain $i_1,i_2\in \left\{1,2,\dots ,{\displaystyle \frac{p-1}{2}}\right\}$, we have

$$|{\psi }_p(i_1,j)|=|{\psi }_p(i_2,j)|.$$

If the numbers ${\sigma }_p(i_1,j)$ and ${\sigma }_p(i_2,j)$ have the same parity, we get

$${mod}_p\left(2i_1{inv}_{p}j\right)={mod}_p\left(2i_2{inv}_{p}j\right).$$

Thus, there exist $r,s\in \mathbb{Z}$ such that

$$2i_1{inv}_{p}j+rp=2i_2{inv}_{p}j+sp\iff 2(i_1-i_2){inv}_{p}j=(s-r)p,$$

where $i_1-i_2\in \left\{0,\pm 1,\pm 2,\dots ,\pm {\displaystyle \frac{p-3}{2}}\right\}$. Because $(s-r)p$ is divisible by p, it follows that $2(i_1-i_2){inv}_{p}j$ must also be divisible by p, which implies that $i_1=i_2$. Conversely, if ${\sigma }_p(i_1,j)$ and ${\sigma }_p(i_2,j)$ have different parity, we obtain

$${mod}_p\left(2i_1{inv}_{p}j\right)+{mod}_p\left(2i_2{inv}_{p}j\right)=p.$$

Thus, there exist $t,u\in \mathbb{Z}$ such that

$$2i_1{inv}_{p}j+tp+2i_2{inv}_{p}j+up=p\iff 2(i_1+i_2){inv}_{p}j=(1-t-u)p,$$

where $i_1+i_2\in \{2,3,\dots ,p-1\}$. This leads to a contradiction, because $(1-t-u)p$ is divisible by p, while $2(i_1+i_2){inv}_{p}j$ is not.

Next, let $i\in \left\{1,2,\dots ,{\displaystyle \frac{p-1}{2}}\right\}$ be fixed, and assume for certain $j_1,j_2\in \left\{1,2,\dots ,{\displaystyle \frac{p-1}{2}}\right\}$ that

$$|{\psi }_p(i,j_1)|=|{\psi }_p(i,j_2)|.$$

If ${\sigma }_p(i,j_1)$ and ${\sigma }_p(i,j_2)$ have the same parity, we get

$${mod}_p\left(2i{inv}_p{j}_1\right)={mod}_p\left(2i{inv}_p{j}_2\right).$$

Hence, there exist $r,s\in \mathbb{Z}$ such that

$$2i{inv}_p{j}_1+rp=2i{inv}_p{j}_2+sp\iff 2i({inv}_p{j}_1-{inv}_p{j}_2)=(s-r)p,$$

where ${inv}_p{j}_1-{inv}_p{j}_2\in \left\{0,\pm 1,\pm 2,\dots ,\pm (p-2)\right\}$. Because $(s-r)p$ is divisible by p, then $2i({inv}_p{j}_1-{inv}_p{j}_2)$ must also be divisible by p, leading to ${inv}_p{j}_1={inv}_p{j}_2$. By the injectivity of the function ${inv}_p$, we get $j_1=j_2$. However, if ${\sigma }_p(i,j_1)$ and ${\sigma }_p(i,j_2)$ have different parity, then

$${mod}_p\left(2i{inv}_p{j}_1\right)+{mod}_p\left(2i{inv}_p{j}_2\right)=p.$$

Thus, there exist $t,u,v,w\in \mathbb{Z}$ such that

$$\begin{array}{cc}\hfill & 2i{inv}_p{j}_1+tp+2i{inv}_p{j}_2+up=p\hfill \\ \hfill & \iff 2i({inv}_p{j}_1+{inv}_p{j}_2)=(1-t-u)p\hfill \\ \hfill & \iff 2i(j_1{j}_2{inv}_p{j}_2+j_2{j}_1{inv}_p{j}_1)=j_1{j}_2(1-t-u)p\hfill \\ \hfill & \iff 2i\left(j_1(1+vp)+j_2(1+wp)\right)=j_1{j}_2(1-t-u)p\hfill \\ \hfill & \iff 2i(j_1+j_2)=\left(j_1{j}_2(1-t-u)-2i(j_{1}v+j_{2}w)\right)p,\hfill \end{array}$$

where $j_1+j_2\in \{2,3,\dots ,p-1\}$. This scenario also leads to a contradiction, because $\left(j_1{j}_2(1-t-u)-2i(j_{1}v+j_{2}w)\right)p$ is divisible by p, while $2i(j_1+j_2)$ is not divisible by p. □

The proofs of the theorems will involve two lemmas, which follow directly from [7].

Lemma 1. 

Let $p\in \mathbb{P}$, $p\ge 3$. Then, the following holds:

$$det{S}_p={(-1)}^{{\displaystyle \frac{(p-1)(p-3)}{8}}}{\left({\displaystyle \frac{p}{4}}\right)}^{{\displaystyle \frac{p-1}{4}}}.$$

Lemma 2. 

Let $p\in \mathbb{P}$, $p\ge 3$. Then, the following holds:

$$S_p^{-1}={\displaystyle \frac{4}{p}}{S}_p.$$

Additionally, we will use the following two trigonometric identities in our proof of the theorems.

Lemma 3. 

Let $p\in \mathbb{P}$, $p\ge 3$. Then, the following identities are satisfied:

$$\sum _{k=1}^{{\displaystyle \frac{p-1}{2}}}{(-1)}^{k+1}sin{\displaystyle \frac{2kn\pi }{p}}={\displaystyle \frac{1}{2}}tan{\displaystyle \frac{n\pi }{p}}\mathit{for}n\in \mathbb{Z}$$

(2)

and

$$\sum _{k=1}^{{\displaystyle \frac{p-1}{2}}}(p-2k)sin{\displaystyle \frac{2kn\pi }{p}}={\displaystyle \frac{p}{2}}cot{\displaystyle \frac{n\pi }{p}}\mathit{for}n\in \mathbb{Z}\setminus p\mathbb{Z}.$$

(3)

Proof. 

Using the formula for the product of sine and cosine, we have

$$\begin{array}{cc}\hfill 2cos{\displaystyle \frac{n\pi }{p}}\sum _{k=1}^{{\displaystyle \frac{p-1}{2}}}{(-1)}^{k+1}sin{\displaystyle \frac{2kn\pi }{p}}& =\sum _{k=1}^{{\displaystyle \frac{p-1}{2}}}{(-1)}^{k+1}\left(sin{\displaystyle \frac{(2k+1)n\pi }{p}}+sin{\displaystyle \frac{(2k-1)n\pi }{p}}\right)\hfill \\ \hfill & =\sum _{k=2}^{{\displaystyle \frac{p+1}{2}}}{(-1)}^{k}sin{\displaystyle \frac{(2k-1)n\pi }{p}}-\sum _{k=1}^{{\displaystyle \frac{p-1}{2}}}{(-1)}^{k}sin{\displaystyle \frac{(2k-1)n\pi }{p}}\hfill \\ \hfill & ={(-1)}^{{\displaystyle \frac{p+1}{2}}}sin\left(n\pi \right)+sin{\displaystyle \frac{n\pi }{p}}=sin{\displaystyle \frac{n\pi }{p}},\hfill \end{array}$$

which proves (2). Next, using the formula for the product of sine and square sine, namely

$$4sinx{sin}^{2}y=-sin(x-2y)+2sinx-sin(x+2y),$$

we obtain

$$\begin{array}{cc}\hfill & 4{sin}^2{\displaystyle \frac{n\pi }{p}}\sum _{k=1}^{{\displaystyle \frac{p-1}{2}}}(p-2k)sin{\displaystyle \frac{2kn\pi }{p}}\hfill \\ \hfill & =\sum _{k=1}^{{\displaystyle \frac{p-1}{2}}}(p-2k)\left(-sin{\displaystyle \frac{(2k-2)n\pi }{p}}+2sin{\displaystyle \frac{2kn\pi }{p}}-sin{\displaystyle \frac{(2k+2)n\pi }{p}}\right)\hfill \\ \hfill & =-\sum _{k=0}^{{\displaystyle \frac{p-3}{2}}}(p-2k-2)sin{\displaystyle \frac{2kn\pi }{p}}+\sum _{k=1}^{{\displaystyle \frac{p-1}{2}}}2(p-2k)sin{\displaystyle \frac{2kn\pi }{p}}-\sum _{k=2}^{{\displaystyle \frac{p+1}{2}}}(p-2k+2)sin{\displaystyle \frac{2kn\pi }{p}}\hfill \\ \hfill & =-(p-2)sin0-(p-4)sin{\displaystyle \frac{2n\pi }{p}}+2(p-2)sin{\displaystyle \frac{2n\pi }{p}}\hfill \\ \hfill & +2sin{\displaystyle \frac{(p-1)n\pi }{p}}-3sin{\displaystyle \frac{(p-1)n\pi }{p}}-sin{\displaystyle \frac{(p+1)n\pi }{p}}\hfill \\ \hfill & =psin{\displaystyle \frac{2n\pi }{p}}-sin{\displaystyle \frac{(p-1)n\pi }{p}}-sin{\displaystyle \frac{(p+1)n\pi }{p}}\hfill \\ \hfill & =2psin{\displaystyle \frac{n\pi }{p}}cos{\displaystyle \frac{n\pi }{p}}-2sin\left(n\pi \right)cos{\displaystyle \frac{n\pi }{p}}=2psin{\displaystyle \frac{n\pi }{p}}cos{\displaystyle \frac{n\pi }{p}},\hfill \end{array}$$

which proves (3). □

Remark 10. 

Formula (3) was already proven in [7] using a different method. We present an alternative proof here due to its other assumptions and compatibility with the proof of (2).

Remark 11. 

Lemmas 1 and 2 remain true for all integers $p\ge 3$ if $S_p$ is defined as mentioned in Remark 1. Moreover, Lemma 3 remains valid for all even integers $p\ge 3$ under the assumption that $n\in \mathbb{Z}\setminus p\left(\mathbb{Z}+{\displaystyle \frac{1}{2}}\right)$ in (2) and $n\in \mathbb{Z}\setminus p\mathbb{Z}$ in (3).

Based on the auxiliary results presented above, we are now ready to prove our main theorems.

Proof of Theorem 3. 

From the definitions of the matrices ${\mathsf{\Phi }}_p$ and ${\mathsf{\Psi }}_p$ and from Lemma 2, we have

$$T_p=S_p^{-1}{S}_p{T}_p=2S_p·{\displaystyle \frac{2}{p}}{S}_p{T}_p=2S_p{\mathsf{\Phi }}_p$$

and

$$C_p=S_p^{-1}{S}_p{C}_p={\displaystyle \frac{2}{p}}{S}_p·2S_p{C}_p={\displaystyle \frac{2}{p}}{S}_p{\mathsf{\Psi }}_p,$$

which confirms the conclusion of the theorem. □

Proof of Theorem 2. 

We define the auxiliary mappings ${\epsilon }_p:{\left\{1,2,\dots ,{\displaystyle \frac{p-1}{2}}\right\}}^2\to \{\pm 1\}$ and ${\tau }_p:{\left\{1,2,\dots ,{\displaystyle \frac{p-1}{2}}\right\}}^2\to \{1,3,\dots ,p-2\}$ as

$${\epsilon }_p(i,j)=\left\{\begin{array}{cc}1,\hfill & \mathrm{if}{\sigma }_p(i,j)\mathrm{is}\mathrm{even},\hfill \\ -1,\hfill & \mathrm{if}{\sigma }_p(i,j)\mathrm{is}\mathrm{odd},\hfill \end{array}\right.$$

and

$${\tau }_p(i,j)=\left|{\psi }_p(i,j)\right|=\left\{\begin{array}{cc}p-{\sigma }_p(i,j),\hfill & \mathrm{if}{\sigma }_p(i,j)\mathrm{is}\mathrm{even},\hfill \\ {\sigma }_p(i,j),\hfill & \mathrm{if}{\sigma }_p(i,j)\mathrm{is}\mathrm{odd}.\hfill \end{array}\right.$$

Moreover, we assert that for any $i,j\in \left\{1,2,\dots ,{\displaystyle \frac{p-1}{2}}\right\}$ the following formulas hold:

$${\phi }_p(i,j)={\epsilon }_p(i,j){(-1)}^{{\displaystyle \frac{p-{\tau }_p(i,j)}{2}}+1}\mathrm{and}{\psi }_p(i,j)={\epsilon }_p(i,j){\tau }_p(i,j),$$

(4)

where, in proving the first of these formulas, we use

$${(-1)}^{{\displaystyle \frac{p-{\tau }_p(i,j)}{2}}+1}=-{(-1)}^{{\displaystyle \frac{p-{\tau }_p(i,j)}{2}}}={(-1)}^p{(-1)}^{-{\displaystyle \frac{p-{\tau }_p(i,j)}{2}}}={(-1)}^{{\displaystyle \frac{p+{\tau }_p(i,j)}{2}}}.$$

We fix $i,j\in \left\{1,2,\dots ,{\displaystyle \frac{p-1}{2}}\right\}$. Because, by Theorem 1, the mapping

$$\left\{1,2,\dots ,{\displaystyle \frac{p-1}{2}}\right\}\ni k\mapsto {\tau }_p(k,j)\in \{1,3,\dots ,p-2\}$$

(5)

is a bijection, we infer that the mapping

$$\left\{1,2,\dots ,{\displaystyle \frac{p-1}{2}}\right\}\ni k\mapsto {\displaystyle \frac{p-{\tau }_p(k,j)}{2}}\in \left\{1,2,\dots ,{\displaystyle \frac{p-1}{2}}\right\}$$

is also a bijection, meaning that it is a permutation of the set $\left\{1,2,\dots ,{\displaystyle \frac{p-1}{2}}\right\}$. Hence, by Lemma 3, assuming $n:=ij$, and noting that $ij\notin p\mathbb{Z}$, we obtain

$$\sum _{k=1}^{{\displaystyle \frac{p-1}{2}}}{(-1)}^{{\displaystyle \frac{p-{\tau }_p(k,j)}{2}}+1}sin{\displaystyle \frac{ij\left(p-{\tau }_p(k,j)\right)\pi }{p}}={\displaystyle \frac{1}{2}}tan{\displaystyle \frac{ij\pi }{p}}$$

(6)

and

$$\sum _{k=1}^{{\displaystyle \frac{p-1}{2}}}{\tau }_p(k,j)sin{\displaystyle \frac{ij\left(p-{\tau }_p(k,j)\right)\pi }{p}}={\displaystyle \frac{p}{2}}cot{\displaystyle \frac{ij\pi }{p}}.$$

(7)

Next, fix $k\in \left\{1,2,\dots ,{\displaystyle \frac{p-1}{2}}\right\}$. If ${\sigma }_p(k,j)$ is even, then there exist $\alpha ,\beta \in \mathbb{Z}$ such that

$$\begin{array}{cc}\hfill jp-j{\tau }_p(k,j)-2k& =jp-j\left(p-{mod}_p\left(2k{inv}_{p}j\right)\right)-2k\hfill \\ \hfill & =jp-j\left(p-(2k{inv}_{p}j+\alpha p)\right)-2k\hfill \\ \hfill & =jp-jp+2kj{inv}_{p}j+j\alpha p-2k\hfill \\ \hfill & =2k(1+\beta p)+j\alpha p-2k=(2k\beta +j\alpha )p.\hfill \end{array}$$

The evenness of the number $jp-j{\tau }_p(k,j)-2k$ implies that ${\displaystyle \frac{jp-j{\tau }_p(k,j)-2k}{2p}}\in \mathbb{Z}$. Hence, using the difference of sines formula, we get

$$\begin{array}{cc}\hfill & sin{\displaystyle \frac{ij\left(p-{\tau }_p(k,j)\right)\pi }{p}}-{\epsilon }_p(k,j)sin{\displaystyle \frac{2ik\pi }{p}}\hfill \\ \hfill & =2sin\left({\displaystyle \frac{jp-j{\tau }_p(k,j)-2k}{2p}}·i\pi \right)cos\left({\displaystyle \frac{jp-j{\tau }_p(k,j)+2k}{2p}}·i\pi \right)=0.\hfill \end{array}$$

If ${\sigma }_p(k,j)$ is odd, then there exist $\gamma ,\delta \in \mathbb{Z}$ such that

$$\begin{array}{cc}\hfill jp-j{\tau }_p(k,j)+2k& =jp-j{mod}_p\left(2k{inv}_{p}j\right)+2k\hfill \\ \hfill & =jp-j(2k{inv}_{p}j+\gamma p)+2k\hfill \\ \hfill & =jp-2kj{inv}_{p}j-j\gamma p+2k\hfill \\ \hfill & =jp-2k(1+\delta p)-j\gamma p+2k=(j-2k\delta -j\gamma )p.\hfill \end{array}$$

The evenness of the number $jp-j{\tau }_p(k,j)+2k$ implies that ${\displaystyle \frac{jp-j{\tau }_p(k,j)+2k}{2p}}\in \mathbb{Z}$. Therefore, using the sum of sines formula, we get

$$\begin{array}{cc}\hfill & sin{\displaystyle \frac{ij\left(p-{\tau }_p(k,j)\right)\pi }{p}}-{\epsilon }_p(k,j)sin{\displaystyle \frac{2ik\pi }{p}}\hfill \\ \hfill & =2sin\left({\displaystyle \frac{jp-j{\tau }_p(k,j)+2k}{2p}}·i\pi \right)cos\left({\displaystyle \frac{jp-j{\tau }_p(k,j)-2k}{2p}}·i\pi \right)=0.\hfill \end{array}$$

In summary, for each $i,j,k\in \left\{1,2,\dots ,{\displaystyle \frac{p-1}{2}}\right\}$, we have

$$sin{\displaystyle \frac{ij\left(p-{\tau }_p(k,j)\right)\pi }{p}}={\epsilon }_p(k,j)sin{\displaystyle \frac{2ik\pi }{p}}.$$

Thus, using (4) and the above identity, Equations (6) and (7) take respectively the forms

$$\sum _{k=1}^{{\displaystyle \frac{p-1}{2}}}{\phi }_p(k,j)sin{\displaystyle \frac{2ik\pi }{p}}={\displaystyle \frac{1}{2}}tan{\displaystyle \frac{ij\pi }{p}}$$

and

$$\sum _{k=1}^{{\displaystyle \frac{p-1}{2}}}{\psi }_p(k,j)sin{\displaystyle \frac{2ik\pi }{p}}={\displaystyle \frac{p}{2}}cot{\displaystyle \frac{ij\pi }{p}}$$

for any $i,j\in \left\{1,2,\dots ,{\displaystyle \frac{p-1}{2}}\right\}$. Equivalently, the following identities hold:

$$T_p=2S_p{\left[{\phi }_p(i,j)\right]}_{{\displaystyle \frac{p-1}{2}}\times {\displaystyle \frac{p-1}{2}}}\mathrm{and}{C}_p={\displaystyle \frac{2}{p}}{S}_p{\left[{\psi }_p(i,j)\right]}_{{\displaystyle \frac{p-1}{2}}\times {\displaystyle \frac{p-1}{2}}}.$$

From Theorem 3, these yield

$$2S_p{\mathsf{\Phi }}_p=2S_p{\left[{\phi }_p(i,j)\right]}_{{\displaystyle \frac{p-1}{2}}\times {\displaystyle \frac{p-1}{2}}}\mathrm{and}{\displaystyle \frac{2}{p}}{S}_p{\mathsf{\Psi }}_p={\displaystyle \frac{2}{p}}{S}_p{\left[{\psi }_p(i,j)\right]}_{{\displaystyle \frac{p-1}{2}}\times {\displaystyle \frac{p-1}{2}}}.$$

Because the matrix $S_p$ is invertible by Lemma 2, it suffices to conclude the proof. □

Proof of Theorem 4. 

Based on Theorem 3 and the properties of the determinant, we have

$$det{T}_p=2^{{\displaystyle \frac{p-1}{2}}}det{S}_{p}det{\mathsf{\Phi }}_p$$

and

$$det{C}_p={\left({\displaystyle \frac{2}{p}}\right)}^{{\displaystyle \frac{p-1}{2}}}det{S}_{p}det{\mathsf{\Psi }}_p.$$

Using Lemma 1, we obtain

$$det{T}_p=2^{{\displaystyle \frac{p-1}{2}}}{(-1)}^{{\displaystyle \frac{(p-1)(p-3)}{8}}}{\left({\displaystyle \frac{p}{4}}\right)}^{{\displaystyle \frac{p-1}{4}}}det{\mathsf{\Phi }}_p={(-1)}^{{\displaystyle \frac{(p-1)(p-3)}{8}}}{p}^{{\displaystyle \frac{p-1}{4}}}det{\mathsf{\Phi }}_p$$

and

$$det{C}_p={\left({\displaystyle \frac{2}{p}}\right)}^{{\displaystyle \frac{p-1}{2}}}{(-1)}^{{\displaystyle \frac{(p-1)(p-3)}{8}}}{\left({\displaystyle \frac{p}{4}}\right)}^{{\displaystyle \frac{p-1}{4}}}det{\mathsf{\Psi }}_p={(-1)}^{{\displaystyle \frac{(p-1)(p-3)}{8}}}{p}^{{\displaystyle \frac{1-p}{4}}}det{\mathsf{\Psi }}_p,$$

which was to be shown. □

Proof of Theorem 5. 

First, note that $T_n^{*}=T_n$, $C_n^{*}=C_n$, and $S_n^{*}=S_n$. Thus, by Theorem 3 and Lemma 2, it follows that

$$T_p{C}_p=T_p^{*}{C}_p=2{\mathsf{\Phi }}_p^{*}{S}_p^{*}·{\displaystyle \frac{2}{p}}{S}_p{\mathsf{\Psi }}_p={\mathsf{\Phi }}_p^{*}{S}_p{S}_p^{-1}{\mathsf{\Psi }}_p={\mathsf{\Phi }}_p^{*}{\mathsf{\Psi }}_p$$

and

$$C_p{T}_p=C_p^{*}{T}_p={\displaystyle \frac{2}{p}}{\mathsf{\Psi }}_p^{*}{S}_p^{*}·2S_p{\mathsf{\Phi }}_p={\mathsf{\Psi }}_p^{*}{S}_p{S}_p^{-1}{\mathsf{\Phi }}_p={\mathsf{\Psi }}_p^{*}{\mathsf{\Phi }}_p,$$

which was to be proved. □