On the norms of an r-circulant matrix with the generalized k-Horadam numbers

Abstract

In this paper, we present new upper and lower bounds for the spectral norm of an r-circulant matrix $H=C_r(H_{k,0},H_{k,1},H_{k,2},\dots ,H_{k,n-1})$ whose entries are the generalized k-Horadam numbers. Furthermore, we obtain new formulas to calculate the eigenvalues and determinant of the matrix H.

MSC:11B39, 15A60, 15A15.

1 Introduction and preliminaries

The r-circulant matrices [1–3] have been one of the most interesting research areas in the field of computation mathematics. It is known that these matrices have a wide range of applications in signal processing, coding theory, image processing, digital image disposal, linear forecast and design of self-regress.

Although there are many works about special matrices and their norms, we can mainly depict the following ones since they will be needed for our results. In [4], Solak defined the $n\times n$ circulant matrices $A=[a_{ij}]$ and $B=[b_{ij}]$, where $a_{ij}\equiv F_{(mod(j-i,n))}$ and $b_{ij}\equiv L_{(mod(j-i,n))}$, and then investigated the upper and lower bounds for A and B, respectively. In [5], they defined circulant matrices involving k-Lucas and generalized k-Fibonacci numbers and also investigated the upper and lower bounds for the norms of these matrices. Later on, Shen and Cen in [6], found some other upper and lower bounds for the spectral norms of r-circulant matrices in the form $A=C_r(F_0,F_1,\dots ,F_{n-1})$, $B=C_r(L_0,L_1,\dots ,L_{n-1})$, where $F_n$ is the n th Fibonacci number and $L_n$ is the n th Lucas number. In [7], the authors found some other upper and lower bounds for the spectral norms of r-circulant matrices associated with the k-Fibonacci and k-Lucas numbers, respectively. It is clear that the same study about similar subject can also be done for different numbers. For instance, in [8], authors defined a circulant matrix whose entries are the generalized k-Horadam numbers and then computed the spectral norm, eigenvalues and determinant of the matrix.

At this point, we can keep going to present the fundamental material which is about the definition of the generalized k-Horadam sequence ${\{H_{k,n}\}}_{n\in \mathbb{N}}$. In fact, by [9], it was defined as the form

$$H_{k,n+2}=f(k)H_{k,n+1}+g(k)H_{k,n},H_{k,0}=a,H_{k,1}=b(a,b\in \mathbb{R}),$$

(1)

where, $n\ge 0$, $k\in {\mathbb{R}}^{+}$ and $f^2(k)+4g(k)>0$. Obviously, if we choose suitable values on $f(k)$, $g(k)$, a and b for (1), then this sequence reduces to the special all second-order sequences in the literature. For example, by taking $f(k)=g(k)=1$, $a=0$ and $b=1$, the well-known Fibonacci sequence is obtained.

Binet’s formula allows us to express the generalized k-Horadam number in function of the roots α and β of the characteristic equation $x^2-f(k)x-g(k)=0$. Binet’s formula related to the sequence ${\{H_{k,n}\}}_{n\in \mathbb{N}}$ has the form

$$H_{k,n}=\frac{X{\alpha }^n-Y{\beta }^n}{\alpha -\beta },$$

(2)

where $X=b-a\beta$ and $Y=b-a\alpha$.

Lemma 1 [9]

Let the entries of each matrix $X_n=\left(\begin{array}{cc}{H}_{k,n-1}& H_{k,n}\\ H_{k,n}& H_{k,n+1}\end{array}\right)$ be the generalized k-Horadam numbers. For $n\ge 1$, we get

$$|X_n|={(-g(k))}^{n-1}(a^{2}g(k)+abf(k)-b^2).$$

Definition 2 For any given $c_0,c_1,c_2,\dots ,c_{n-1}\in \mathbb{C}$, the r-circulant matrix $C_r={(c_{ij})}_{n\times n}$ is defined by

$$C_r=\left(\begin{array}{cccccc}{c}_0& c_1& c_2& \cdots & c_{n-2}& c_{n-1}\\ r{c}_{n-1}& c_0& c_1& \cdots & c_{n-3}& c_{n-2}\\ r{c}_{n-2}& r{c}_{n-1}& c_0& \cdots & c_{n-4}& c_{n-3}\\ ⋮& ⋮& ⋮& \ddots & ⋮& ⋮\\ r{c}_2& r{c}_3& r{c}_4& \cdots & c_0& c_1\\ r{c}_1& r{c}_2& r{c}_3& \cdots & r{c}_{n-1}& c_0\end{array}\right).$$

Let $C_r=(c_0,c_1,c_2,\dots ,c_{n-1})$ denote an r-circulant matrix.

It is obvious that the matrix $C_r$ turns into a classical circulant matrix for $r=1$. Let us take any $A=[a_{ij}]\in M_{n,n}(\mathbb{C})$ (which could be a circulant matrix as well). The well-known Frobenius (or Euclidean) norm of the matrix A is given by

$${\parallel A\parallel }_F={[\sum _{i=1}^m\sum _{j=1}^n|a_{ij}{|}^2]}^{\frac{1}{2}},$$

and also the spectral norm of A is presented by

$${\parallel A\parallel }_2=\sqrt{\underset{1\le i\le n}{max}{\lambda }_i\left(A^{H}A\right)},$$

where ${\lambda }_i(A^{H}A)$ are the eigenvalues of $A^{H}A$ such that $A^H$ is the conjugate transpose of A. Then it is quite well known that

$$\frac{1}{\sqrt{n}}{\parallel A\parallel }_F\le {\parallel A\parallel }_2\le {\parallel A\parallel }_F.$$

The following lemma will be needed in the proof of Theorem 5 below.

Lemma 3 [10]

For any matrices $A=[a_{ij}]\in M_{m,n}(\mathbb{C})$ and $B=[b_{ij}]\in M_{m,n}(\mathbb{C})$, we have

$${\parallel A\circ B\parallel }_2\le r_1(A)c_1(B),$$

where $A\circ B$ is the Hadamard product and $r_1(A)={max}_{1\le i\le n}\sqrt{{\sum }_{j=1}^n|a_{ij}{|}^2}$, $c_1(B)={max}_{1\le j\le n}\sqrt{{\sum }_{i=1}^n|b_{ij}{|}^2}$.

Throughout this paper, the r-circulant matrix, whose entries are the generalized k-Horadam numbers, will be denoted by $H=C_r(H_{k,0},H_{k,1},\dots ,H_{k,n-1})$.

In this paper, we first give lower and upper bounds for the spectral norms of H. In particular, by taking $r=1$, we obtain lower and upper bounds for the spectral norms of the circulant matrix associated with the generalized k-Horadam numbers. Afterwards, we also formulate the eigenvalues and determinant of the matrix H defined above.

2 Main results

Let us first consider the following lemma which states the sum square of the generalized k-Horadam numbers.

Lemma 4 For $n\ge 1$, we have

$$\sum _{i=0}^{n-1}{H}_{k,i}^2=\frac{H_{k,n}^2-g^2(k)H_{k,n-1}^2+M}{f^2(k)+2g(k)-g^2(k)-1},$$

where $M={(b-af(k))}^2-a^2-2XY(\frac{1-{(-g(k))}^n}{1+g(k)})$, $X=b-a\beta$ and $Y=b-a\alpha$.

Proof Let $A={\sum }_{i=0}^{n-1}{H}_{k,i}^2$. By using (1), we have

$$\begin{array}{rcl}A& =& \sum _{i=0}^{n-1}{\left(\frac{H_{k,i+1}-g(k)H_{k,i-1}}{f(k)}\right)}^2\\ =& \frac{{\sum }_{i=0}^{n-1}{H}_{k,i+1}^2+g^2(k){\sum }_{i=0}^{n-1}{H}_{k,i-1}^2-2g(k){\sum }_{i=0}^{n-1}{H}_{k,i+1}{H}_{k,i-1}}{f^2(k)}.\end{array}$$

By considering (1), we obtain

$$\begin{array}{rcl}A{f}^2(k)& =& A(1+g^2(k)-2g(k))+H_{k,n}^2-g^2(k)H_{k,n-1}-a^2\\ +{(b-af(k))}^2-2XY\left(\frac{1-{(-g(k))}^n}{1+g(k)}\right)\end{array}$$

which is desired. □

The following theorem gives us the upper and lower bounds for the spectral norms of the matrix H.

Theorem 5 Let $H=C_r(H_{k,0},H_{k,1},H_{k,2},\dots ,H_{k,n-1})$ be an r-circulant matrix. Then we have

if $|r|\ge 1$, then

$$\sqrt{\sum _{i=0}^{n-1}{H}_{k,i}^2}\le {\parallel H\parallel }_2\le \sqrt{(a^2(1-|r{|}^2)+|r{|}^2\sum _{i=0}^{n-1}{H}_{k,i}^2)(1-a^2+\sum _{i=0}^{n-1}{H}_{k,i}^2)},$$

if $|r|<1$, then

$$|r|\sqrt{\sum _{i=0}^{n-1}{H}_{k,i}^2}\le {\parallel H\parallel }_2\le \sqrt{n\sum _{i=0}^{n-1}{H}_{k,i}^2},$$

where $r\in \mathbb{C}$, ${\sum }_{i=0}^{n-1}{H}_{k,i}^2=\frac{H_{k,n}^2-g^2(k)H_{k,n-1}^2+M}{f^2(k)+2g(k)-g^2(k)-1}$, $M={(b-af(k))}^2-a^2-2XY(\frac{1-{(-g(k))}^n}{1+g(k)})$.

Proof The matrix H is of the form

$$H=\left[\begin{array}{ccccc}{H}_{k,0}& H_{k,1}& H_{k,2}& \cdots & H_{k,n-1}\\ r{H}_{k,n-1}& H_{k,0}& H_{k,1}& \cdots & H_{k,n-2}\\ r{H}_{k,n-2}& r{H}_{k,n-1}& H_{k,0}& \cdots & H_{k,n-3}\\ ⋮& ⋮& ⋮& \ddots & ⋮\\ r{H}_{k,1}& r{H}_{k,2}& r{H}_{k,3}& \cdots & H_{k,0}\end{array}\right]$$

(3)

and, by the definition of Frobenius norm, we clearly have

$${\parallel H\parallel }_F^2=\sum _{i=0}^{n-1}(n-i)H_{k,i}^2+\sum _{i=1}^{n-1}i|r{|}^2{H}_{k,i}^2.$$

If $|r|\ge 1$, then we obtain

$${\parallel H\parallel }_F^2\ge \sum _{i=0}^{n-1}(n-i)H_{k,i}^2+\sum _{i=1}^{n-1}i{H}_{k,i}^2=n\sum _{i=0}^{n-1}{H}_{k,i}^2.$$

Also, by considering Lemma 4, we can write

$${\parallel H\parallel }_F^2\ge n\left[\frac{H_{k,n}^2-g^2(k)H_{k,n-1}^2+M}{f^2(k)+2g(k)-g^2(k)-1}\right].$$

It follows that

$$\frac{{\parallel H\parallel }_F}{\sqrt{n}}\ge \sqrt{\frac{H_{k,n}^2-g^2(k)H_{k,n-1}^2+M}{f^2(k)+2g(k)-g^2(k)-1}},$$

where $M={(b-af(k))}^2-a^2-2XY(\frac{1-{(-g(k))}^n}{1+g(k)})$. Then by (1), we have

$${\parallel H\parallel }_2\ge \sqrt{\frac{H_{k,n}^2-g^2(k)H_{k,n-1}^2+M}{f^2(k)+2g(k)-g^2(k)-1}}.$$

Similarly, for $|r|<1$, we can write

$$\begin{array}{rcl}{\parallel H\parallel }_F^2& =& \sum _{i=0}^{n-1}(n-i)H_{k,i}^2+\sum _{i=1}^{n-1}i|r{|}^2{H}_{k,i}^2\\ \ge & \sum _{i=0}^{n-1}(n-i)|r{|}^2{H}_{k,i}^2+\sum _{i=1}^{n-1}i|r{|}^2{H}_{k,i}^2=n|r{|}^2\sum _{i=0}^{n-1}{H}_{k,i}^2.\end{array}$$

Again, by considering Lemma 4 and (1), we get

$$\frac{{\parallel H\parallel }_F}{\sqrt{n}}\ge |r|\sqrt{\sum _{i=1}^{n-1}{H}_{k,i}^2}.$$

It follows that

$${\parallel H\parallel }_2\ge |r|\sqrt{\frac{H_{k,n}^2-g^2(k)H_{k,n-1}^2+M}{f^2(k)+2g(k)-g^2(k)-1}}.$$

Now, for $|r|\ge 1$, we give the upper bound for the spectral norm of the matrix H as in the following. Let the matrices B and C be as

$$\begin{array}{c}B=\left(\begin{array}{ccccc}{H}_{k,0}& 1& 1& \cdots & 1\\ r{H}_{k,n-1}& H_{k,0}& 1& \cdots & 1\\ r{H}_{k,n-2}& r{H}_{k,n-1}& H_{k,0}& \cdots & 1\\ ⋮& ⋮& ⋮& \ddots & ⋮\\ r{H}_{k,1}& r{H}_{k,2}& r{H}_{k,3}& \cdots & H_{k,0}\end{array}\right)\text{and}\hfill \\ C=\left(\begin{array}{ccccc}1& H_{k,1}& H_{k,2}& \cdots & H_{k,n-1}\\ 1& 1& H_{k,1}& \cdots & H_{k,n-2}\\ 1& 1& 1& \cdots & H_{k,n-3}\\ ⋮& ⋮& ⋮& \ddots & ⋮\\ 1& 1& 1& \cdots & 1\end{array}\right)\hfill \end{array}$$

such that $H=B\circ C$. Then we obtain

$$\begin{array}{r}\begin{array}{rl}{r}_1(B)& =\underset{1\le i\le n}{max}\sqrt{\sum _{j=1}^n|b_{ij}{|}^2}=\sqrt{H_{k,0}^2+|r{|}^2\sum _{j=1}^{n-1}{H}_{k,j}^2}\\ =\sqrt{a^2(1-|r{|}^2)+|r{|}^2\frac{H_{k,n}^2-g^2(k)H_{k,n-1}^2+M}{f^2(k)+2g(k)-g^2(k)-1}},\end{array}\\ \begin{array}{rl}{c}_1(C)& =\underset{1\le j\le n}{max}\sqrt{\sum _{i=1}^n|c_{ij}{|}^2}=\sqrt{1+\sum _{i=1}^{n-1}{H}_{k,i}^2}\\ =\sqrt{1-a^2+\frac{H_{k,n}^2-g^2(k)H_{k,n-1}^2+M}{f^2(k)+2g(k)-g^2(k)-1}},\end{array}\end{array}$$

where $M={(b-af(k))}^2-a^2-2XY(\frac{1-{(-g(k))}^n}{1+g(k)})$, $X=b-a\beta$ and $Y=b-a\alpha$. By considering Lemma 3, we can write

$${\parallel H\parallel }_2\le r_1(B)\circ c_1(C)=\sqrt{a^2(1-|r{|}^2)+|r{|}^2\sum _{i=0}^{n-1}{H}_{k,i}^2}\sqrt{1-a^2+\sum _{i=0}^{n-1}{H}_{k,i}^2}.$$

For $|r|<1$, we give the upper bound for the spectral norm of the matrix H as in the following. On the other hand, if the matrices D and E are

$$\begin{array}{c}D=\left(\begin{array}{ccccc}1& 1& 1& \cdots & 1\\ r& 1& 1& \cdots & 1\\ r& r& 1& \cdots & 1\\ ⋮& ⋮& ⋮& \ddots & ⋮\\ r& r& r& \cdots & 1\end{array}\right)\text{and}\hfill \\ E=\left(\begin{array}{ccccc}{H}_{k,0}& H_{k,1}& H_{k,2}& \cdots & H_{k,n-1}\\ H_{k,n-1}& H_{k,0}& H_{k,1}& \cdots & H_{k,n-2}\\ H_{k,n-2}& H_{k,n-1}& H_{k,0}& \cdots & H_{k,n-3}\\ ⋮& ⋮& ⋮& \ddots & ⋮\\ H_{k,1}& H_{k,2}& H_{k,3}& \cdots & H_{k,0}\end{array}\right)\hfill \end{array}$$

such that $H=D\circ E$, then we obtain

$$\begin{array}{c}{r}_1(D)=\underset{1\le i\le n}{max}\sqrt{\sum _{j=1}^n|d_{ij}{|}^2}=\sqrt{n},\hfill \\ c_1(E)=\underset{1\le j\le n}{max}\sqrt{\sum _{i=1}^n|e_{ij}{|}^2}=\sqrt{\sum _{i=0}^{n-1}{H}_{k,i}^2}=\sqrt{\frac{H_{k,n}^2-g^2(k)H_{k,n-1}^2+M}{f^2(k)+2g(k)-g^2(k)-1}},\hfill \end{array}$$

where $M={(b-af(k))}^2-a^2-2XY(\frac{1-{(-g(k))}^n}{1+g(k)})$, $X=b-a\beta$ and $Y=b-a\alpha$. By Lemma 3, then we have

$${\parallel H\parallel }_2\le r_1(D)\circ c_1(E)=\sqrt{n\frac{H_{k,n}^2-g^2(k)H_{k,n-1}^2+M}{f^2(k)+2g(k)-g^2(k)-1}},$$

as required. □

Lemma 6 Let H be an $n\times n$ r-circulant matrix. Then we have

$${\lambda }_j(H)=\sum _{p=0}^{n-1}{a}_p{r}^{\frac{p}{n}}{w}^{-jp},$$

where $w=e^{\frac{2\pi i}{n}}$, $i=\sqrt{-1}$, $j=0,1,\dots ,n-1$.

Considering Lemma 6, we obtain the following theorem which gives us the eigenvalues of the matrix in (3).

Theorem 7 Let $H=C_r(H_{k,0},H_{k,1},H_{k,2},\dots ,H_{k,n-1})$ be an r-circulant matrix. Then the eigenvalues of H are written by

$${\lambda }_j(H)=\frac{r{H}_{k,n}+(g(k)r{H}_{k,n-1}-b+af(k))r^{\frac{1}{n}}{w}^{-j}-H_{k,0}}{g(k)r^{\frac{2}{n}}{w}^{-2j}+f(k)r^{\frac{1}{n}}{w}^{-j}-1},$$

where $H_{k,n}$ is the nth generalized k-Horadam number and $w=e^{\frac{2\pi i}{n}}$, $i=\sqrt{-1}$, $j=0,1,\dots ,n-1$.

Proof By Lemma 6, we have ${\lambda }_j(H)={\sum }_{i=0}^{n-1}{H}_{k,i}{r}^{\frac{i}{n}}{w}^{-ji}$. Moreover, by (2), ${\sum }_{i=0}^{n-1}{H}_{k,i}{r}^{\frac{i}{n}}{w}^{-ji}={\sum }_{i=0}^{n-1}\frac{X{\alpha }^n-Y{\beta }^n}{\alpha -\beta }{r}^{\frac{i}{n}}{w}^{-ji}$. Hence, we obtain

$$\begin{array}{rcl}\sum _{i=0}^{n-1}{H}_{k,i}{r}^{\frac{i}{n}}{w}^{-ji}& =& \frac{1}{\alpha -\beta }[X\sum _{i=0}^{n-1}{\left(\alpha r^{\frac{1}{n}}{w}^{-j}\right)}^i-Y\sum _{i=0}^{n-1}{\left(\beta r^{\frac{1}{n}}{w}^{-j}\right)}^i]\\ =& \frac{1}{\alpha -\beta }[\frac{X[{(\alpha r^{\frac{1}{n}}{w}^{-j})}^n-1]}{\alpha r^{\frac{1}{n}}{w}^{-j}-1}-\frac{Y[{(\beta r^{\frac{1}{n}}{w}^{-j})}^n-1]}{\beta r^{\frac{1}{n}}{w}^{-j}-1}]\\ =& \frac{r{H}_{k,n}+(g(k)r{H}_{k,n-1}-b+af(k))r^{\frac{1}{n}}{w}^{-j}-H_{k,0}}{g(k)r^{\frac{2}{n}}{w}^{-2j}+f(k)r^{\frac{1}{n}}{w}^{-j}-1}\end{array}$$

which completes the proof. □

Theorem 8 The determinant of $H=C_r(H_{k,0},H_{k,1},H_{k,2},\dots ,H_{k,n-1})$ is formulated by

$$det(H)=\frac{{(H_{k,0}-r{H}_{k,n})}^n-{(g(k)r{H}_{k,n-1}-b+af(k))}^{n}r}{(1-r{\alpha }^n)(1-r{\beta }^n)}.$$

Proof It is clear that $det(H)={\prod }_{j=0}^{n-1}{\lambda }_j(H)$. By Theorem 7, we get

$$\prod _{j=0}^{n-1}\frac{(H_{k,0}-r{H}_{k,n})-(g(k)r{H}_{k,n-1}-b+af(k))r^{\frac{1}{n}}{w}^{-j}}{(\alpha r^{\frac{1}{n}}{w}^{-j}-1)(\beta r^{\frac{1}{n}}{w}^{-j}-1)}.$$

Also, by considering the well-known identity ${\prod }_{k=0}^{n-1}(x-y{w}^k)=x^n-y^n$, we can write

$$det(H)=\frac{{(H_{k,0}-r{H}_{k,n})}^n-{(g(k)r{H}_{k,n-1}-b+af(k))}^{n}r}{(1-r{\alpha }^n)(1-r{\beta }^n)}.$$

Thus the proof is completed, as desired. □

Remark 9 We should note that choosing suitable values on r, $f(k)$, $g(k)$, a and b in Theorems 5, 7 and 8, lower and upper bounds of the spectral norm, eigenvalues and determinant of r-circulant matrices for the special all second-order sequences are actually obtained.