Realization of Extremal Spectral Data by Pentadiagonal Matrices

Abstract

In this paper, we address the extremal inverse eigenvalue problem for pentadiagonal matrices. We provide sufficient conditions for their existence and realizability through new constructions that consider spectral data of its leading principal submatrices. Finally, we present some examples generated from the algorithmic procedures derived from our results.

1. Introduction

This paper is concerned with the inverse extremal eigenvalues problem for pentadiagonal matrices. An $n\times n$ matrix $\mathcal{A}=\left({\mathbf{a}}_{ij}\right)$ is called pentadiagonal if ${\mathbf{a}}_{ij}=0$ for $|i-j|>2$, where $n\ge 5$ is required. In particular, we consider real symmetric pentadiagonal matrices of the form

$$\mathcal{P}=\left(\begin{array}{ccccccc}{a}_1& b_1& c_1& 0& 0& \dots & 0\\ b_1& a_2& b_2& c_2& 0& \dots & 0\\ c_1& b_2& a_3& \ddots & \ddots & \ddots & ⋮\\ 0& c_2& \ddots & \ddots & \ddots & \ddots & 0\\ 0& 0& \ddots & \ddots & \ddots & b_{n-2}& c_{n-2}\\ ⋮& ⋮& \ddots & \ddots & b_{n-2}& a_{n-1}& b_{n-1}\\ 0& 0& \dots & 0& c_{n-2}& b_{n-1}& a_n\end{array}\right),$$

(1)

with $c_i>0$ and $b_i>0$, and real nonsymmetric pentadiagonal matrices

$$\tilde{\mathcal{P}}=\left(\begin{array}{ccccccc}{a}_1& b_1& d_1& 0& 0& \dots & 0\\ c_1& a_2& b_2& d_2& 0& \dots & 0\\ e_1& c_2& a_3& \ddots & \ddots & \ddots & ⋮\\ 0& e_2& \ddots & \ddots & \ddots & \ddots & 0\\ 0& 0& \ddots & \ddots & \ddots & b_{n-2}& d_{n-2}\\ ⋮& ⋮& \ddots & \ddots & c_{n-2}& a_{n-1}& b_{n-1}\\ 0& 0& \dots & 0& e_{n-2}& c_{n-1}& a_n\end{array}\right).$$

(2)

The extremal inverse eigenvalues problem consists of determining necessary and/or sufficient conditions for the existence and realization of a structured matrix from extremal spectral data of its leading principal submatrices, which can be the minimal or maximal eigenvalues of the leading principal submatrices and some eigenvectors associated with them. In this regard, several papers discuss this problem for different structured matrices and some applications [1,2,3,4,5,6,7,8]. In most cases, the reconstructed matrices are symmetric, and as known, these matrices have real eigenvalues (see [9]). This problem differs from the classical inverse eigenvalues problem in which spectral information of the entire matrix is usually considered only (see [10,11]).

Pentadiagonal matrices often arise in numerical analysis. In particular, it is used to solve the inverse problem of a vibrating beam [12,13,14]. In the literature, there are different works on calculating the inverse matrix, the determinant, the characteristic polynomial, the eigenvalues, and the eigenvectors of a pentadiagonal matrix [15,16,17,18,19]. However, most consider special pentadiagonal matrices, such as Toeplitz and Companion, or with some null entries on their first or second diagonal [20,21,22,23,24]. Boley and Golub were the first to study the inverse eigenvalue problem for these matrices. In [25], they propose the construction of pentadiagonal symmetric matrices by extending the result obtained by Hochstadt [26] for tridiagonal symmetric matrices, using the eigenvalues of three principal submatrices and the block Lanczos algorithm. Ghanbari, in [27], also considers three spectra associated with blocks of a pentadiagonal matrix. The procedure used is similar to that given by Gladwell in [13] to construct tridiagonal matrices partitioned by blocks. Li presents another procedure in [28], where a pentadiagonal matrix is reconstructed from the eigenvalues of the matrix, but has as the leading principal submatrix a given pentadiagonal matrix. Wang gives a very different proposal in [29], where they use three autopairs for the construction of a symmetric pentadiagonal matrix. In the nonsymmetric case, there are very few advances, as we know, only for H-symmetric pentadiagonal matrices [30].

Throughout this paper, we denote by ${\mathcal{A}}_j;j=1,2,\dots ,n$, the leading principal submatrix of a matrix $\mathcal{A}$ of the order n. The eigenvalues of ${\mathcal{A}}_j$ are denoted and arranged in increasing order: ${\lambda }_1^{\left(j\right)}\le {\lambda }_2^{\left(j\right)}\le \dots \le {\lambda }_j^{\left(j\right)}$. $P_j\left(\lambda \right)=det\left(\lambda I_j-{\mathcal{A}}_j\right)$ denotes the characteristic polynomial of the submatrix ${\mathcal{A}}_j$, where $I_j$ is the identity matrix of order j. The minimal eigenvalue ${\lambda }_1^{\left(j\right)}$ and maximal eigenvalue ${\lambda }_j^{\left(j\right)}$ of ${\mathcal{A}}_j$ are called extremal eigenvalues of the leading principal submatrix ${\mathcal{A}}_j$.

We are interested in constructing pentadiagonal matrices of the forms (1) and (2). For this purpose, we propose the following problems:

Problem 1.

Given the set of $2n-1$ real numbers,

$$\left\{{\lambda }_1^{\left(n\right)},\dots ,{\lambda }_1^{\left(j\right)},\dots ,{\lambda }_1^{\left(2\right)},{\lambda }_1^{\left(1\right)},{\lambda }_2^{\left(2\right)},\dots ,{\lambda }_j^{\left(j\right)},\dots ,{\lambda }_n^{\left(n\right)}\right\},$$

and a nonzero vector $\mathbf{x}={(x_1,x_2,\dots ,x_n)}^T$, construct a symmetric pentadiagonal matrix $\mathcal{P}$ of the form (1), such that ${\lambda }_1^{\left(j\right)}$ and ${\lambda }_j^{\left(j\right)}$ are the extremal eigenvalues of the leading principal submatrix ${\mathcal{P}}_j,j=1,2,\dots ,n$ of $\mathcal{P}$, and $\left({\lambda }_n^{\left(n\right)},\mathbf{x}\right)$ is an eigenpair of $\mathcal{P}$.

Problem 2.

Given the set of $3n-3$ real numbers,

$$\left\{{\lambda }_1^{\left(n\right)},\dots ,{\lambda }_1^{\left(j\right)},\dots ,{\lambda }_1^{\left(2\right)},{\lambda }_1^{\left(1\right)},{\lambda }_2^{\left(2\right)},\dots ,{\lambda }_j^{\left(j\right)},\dots ,{\lambda }_n^{\left(n\right)},{\ell }_1,{\ell }_2,\dots ,{\ell }_{n-2}\right\},$$

and the vectors $\mathbf{x}={\left(x_1,\dots ,x_n\right)}^T$ and $\mathbf{y}={\left(y_1,\dots ,y_{n-1}\right)}^T$, construct a nonsymmetric pentadiagonal matrix $\tilde{\mathcal{P}}$ of the form (2) with $a_{i+2}={\ell }_i;i=1,\dots ,n-2$, such that ${\lambda }_1^{\left(j\right)}$ and ${\lambda }_j^{\left(j\right)}$ are the extremal eigenvalues of the leading principal submatrix $\widetilde{{\mathcal{P}}_j}$, $\left({\lambda }_n^{\left(n\right)},\mathbf{x}\right)$ is an eigenpair of the matrix $\tilde{\mathcal{P}}$ and $\left({\lambda }_{n-1}^{(n-1)},\mathbf{y}\right)$ is an eigenpair of the leading principal submatrix ${\tilde{\mathcal{P}}}_{n-1}$.

The following lemmas are fundamental in the proofs of the main results of this work.

Lemma 1

([6]). Let $P\left(\lambda \right)$ be a monic polynomial of degree n with all real zeroes. If ${\lambda }_1$ and ${\lambda }_n$ are, respectively, the smallest and largest zero of $P\left(\lambda \right)$, then

(1) 

If $\mu <{\lambda }_1$, then ${(-1)}^{n}P\left(\mu \right)>0$.

(2) 

If $\mu >{\lambda }_n$, then $P\left(\mu \right)>0$.

Lemma 2

(Theorem 10.1.1., [31]). (Cauchy’s interlacing theorem) Let ${\lambda }_1\le {\lambda }_2\le \dots \le {\lambda }_n$ be the eigenvalues of an $n\times n$ real symmetric matrix A and ${\mu }_1\le {\mu }_2\le \dots \le {\mu }_{n-1}$ be the eigenvalues of an $(n-1)\times (n-1)$ principal submatrix of A, then

$${\lambda }_1\le {\mu }_1\le {\lambda }_2\le {\mu }_2\le \dots \le {\lambda }_{n-1}\le {\mu }_{n-1}\le {\lambda }_n.$$

An immediate consequence of this lemma is that, for any symmetric matrix, particularly for a pentadiagonal matrix,

$${\lambda }_1^{\left(n\right)}\le \cdots \le {\lambda }_1^{\left(j\right)}\le \cdots \le {\lambda }_1^{\left(2\right)}\le {\lambda }_1^{\left(1\right)}\le {\lambda }_2^{\left(2\right)}\le \cdots \le {\lambda }_j^{\left(j\right)}\le \cdots \le {\lambda }_n^{\left(n\right)}.$$

Lemma 3.

An $n\times n$ matrix nonsymmetric pentadiagonal $\tilde{\mathcal{P}}$ of the form (2) is diagonally similar to a matrix symmetric pentadiagonal $\mathcal{P}$ of the form (1) if $b_i{c}_i>0$, $d_i{e}_i>0$ and

$$\prod _{j=i}^{i+1}\frac{c_j}{b_j}=\frac{e_i}{d_i},i=1,2,\dots ,n-2.$$

(3)

Proof. 

The similarity is followed considering the diagonal matrix $D=diag\left\{{\xi }_1,{\xi }_2,...,{\xi }_n\right\}$, where ${\xi }_i^2={\displaystyle \frac{c_i{c}_{i+1}\cdots c_{n-1}}{b_i{b}_{i+1}\cdots b_{n-1}}};i=1,2,\dots ,n-1$ and ${\xi }_n=1$. Then, $D\tilde{\mathcal{P}}{D}^{-1}$ is a symmetric pentadiagonal matrix of the form (1). □

Note that the above result provides a procedure for generating nonsymmetric pentadiagonal matrices, whose eigenvalues of their leading principal submatrices are real, from a symmetric pentadiagonal matrix. Consequently, the matrices in question have the same extremal eigenvalues.

This paper is organized as follows: In Section 2, we discuss a solution to Problem 1 and provide sufficient conditions for the existence and construction of a pentadiagonal symmetric matrix. In Section 3, we study Problem 2 and obtain also sufficient conditions for the nonsymmetric case. Finally, in Section 4, we show some examples to illustrate the results.

2. Construction of Symmetric Pentadiagonal Matrices

The results in this section are related to the existence of a symmetric pentadiagonal matrix. We provide a new construction of such matrices from the extremal eigenvalues of the leading principal submatrices. We start with the following lemma that is fundamental to the development of our results.

Lemma 4.

Let $\mathcal{P}$ be an $n\times n$ symmetric pentadiagonal matrix of the form (1), and let ${\mathcal{P}}_j$ be the $j\times j$ leading principal submatrix of $\mathcal{P}$ with characteristic polynomial $P_j\left(\lambda \right)=det(\lambda I_j-{\mathcal{P}}_j),$ $j=1,2,\dots ,n$. Then, the sequence ${\left\{P_j\left(\lambda \right)\right\}}_{j=1}^n$ satisfies the recurrence relation:

$$\begin{array}{ccc}\hfill P_1\left(\lambda \right)& =& \lambda -a_1,\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hfill \end{array}$$

(4)

$$\begin{array}{ccc}\hfill P_2\left(\lambda \right)& =& (\lambda -a_2)P_1\left(\lambda \right)-b_1^2,\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hfill \end{array}$$

(5)

$$\begin{array}{ccc}\hfill P_j\left(\lambda \right)& =& (\lambda -a_j)P_{j-1}\left(\lambda \right)-b_{j-1}^2{P}_{j-2}\left(\lambda \right)-c_{j-2}^2{Q}_{j-1}\left(\lambda \right)+2b_{j-1}{c}_{j-2}{R}_{j-1}\left(\lambda \right),\hfill \\ & & j=3,4,\dots ,n,\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hfill \end{array}$$

(6)

where $Q_{j-1}\left(\lambda \right)$ and $R_{j-1}\left(\lambda \right)$ are the determinants of the submatrices resulting from eliminating the $(j-2)$-th row and column, and the $(j-1)$-th row and $(j-2)$-th column of submatrix $\lambda I_{j-1}-{\mathcal{P}}_{j-1}$, respectively.

Proof of Lemma 4.

It is immediate by expanding $det(\lambda I_j-{\mathcal{P}}_j)$. □

Before presenting our main results concerning Problem 1, we define the following notations:

$$\left.\begin{array}{cc}\hfill {\alpha }_j& =P_{j-2}\left({\lambda }_1^{\left(j\right)}\right)P_{j-1}\left({\lambda }_j^{\left(j\right)}\right)-P_{j-2}\left({\lambda }_j^{\left(j\right)}\right)P_{j-1}\left({\lambda }_1^{\left(j\right)}\right)\hfill \\ \hfill {\beta }_j& =R_{j-1}\left({\lambda }_j^{\left(j\right)}\right)P_{j-1}\left({\lambda }_1^{\left(j\right)}\right)-R_{j-1}\left({\lambda }_1^{\left(j\right)}\right)P_{j-1}\left({\lambda }_j^{\left(j\right)}\right)\hfill \\ \hfill {\gamma }_j& =Q_{j-1}\left({\lambda }_1^{\left(j\right)}\right)P_{j-1}\left({\lambda }_j^{\left(j\right)}\right)-Q_{j-1}\left({\lambda }_j^{\left(j\right)}\right)P_{j-1}\left({\lambda }_1^{\left(j\right)}\right)\hfill \\ \hfill {\delta }_j& =\left({\lambda }_j^{\left(j\right)}-{\lambda }_1^{\left(j\right)}\right)P_{j-1}\left({\lambda }_1^{\left(j\right)}\right)P_{j-1}\left({\lambda }_j^{\left(j\right)}\right)\hfill \end{array}\right\}$$

(7)

for $j=3,4,\dots ,n$.

Theorem 1.

Let $2n-1$ be real numbers ${\left\{{\lambda }_1^{\left(j\right)},{\lambda }_j^{\left(j\right)}\right\}}_{j=1}^n$ satisfying

$${\lambda }_1^{\left(n\right)}<\cdots <{\lambda }_1^{\left(j\right)}<\cdots <{\lambda }_1^{\left(2\right)}<{\lambda }_1^{\left(1\right)}<{\lambda }_2^{\left(2\right)}<\cdots <{\lambda }_j^{\left(j\right)}<\cdots <{\lambda }_n^{\left(n\right)}.$$

(8)

If

$${\beta }_j^2-{\alpha }_j{\gamma }_j\ne 0$$

(9)

for $j=3,4,\dots ,n$, where ${\alpha }_j,{\beta }_j$, and ${\gamma }_j$ are as in (7), there exists a symmetric pentadiagonal matrix $\mathcal{P}$ of the form (1), such that ${\lambda }_1^{\left(j\right)}$ and ${\lambda }_j^{\left(j\right)}$ are the extremal eigenvalues of the leading principal submatrix ${\mathcal{P}}_j$ of $\mathcal{P},j=1,2,\dots ,n.$

Proof of Theorem 1.

Suppose that ${\left\{{\lambda }_1^{\left(j\right)},{\lambda }_j^{\left(j\right)}\right\}}_{j=2}^n$ satisfies (8). Determining the existence of a symmetric pentadiagonal matrix A with the required properties is equivalent to proving that the system of equations

$$\begin{array}{ccc}\hfill P_j\left({\lambda }_1^{\left(j\right)}\right)& =& 0\hfill \\ \hfill P_j\left({\lambda }_j^{\left(j\right)}\right)& =& 0,j=1,2,\dots ,n\hfill \end{array}$$

(10)

has real solutions $a_j$, $j=1,2,\dots ,n$, $b_j,$$j=1,2,\dots ,n-1$ and $c_j,$$j=1,2,\dots ,n-2$, where the characteristic polynomials $P_j\left(\lambda \right),$$j=1,2,\dots ,n$ satisfy Lemma 4.

It is clear that, from System (10) for $j=1,2$, and Lemma 1, we can obtain the required the entries $a_1,a_2$ and $b_1$.

Now, from System (10), for $j=3,4,\dots ,n$, we have

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{P}_j\left({\lambda }_1^{\left(j\right)}\right)=({\lambda }_1^{\left(j\right)}-a_j)P_{j-1}\left({\lambda }_1^{\left(j\right)}\right)-b_{j-1}^2{P}_{j-2}\left({\lambda }_1^{\left(j\right)}\right)-c_{j-2}^2{Q}_{j-1}\left({\lambda }_1^{\left(j\right)}\right)\hfill \\ +2b_{j-1}{c}_{j-2}{R}_{j-1}\left({\lambda }_1^{\left(j\right)}\right)=0\hfill \\ P_j\left({\lambda }_j^{\left(j\right)}\right)=({\lambda }_j^{\left(j\right)}-a_j)P_{j-1}\left({\lambda }_j^{\left(j\right)}\right)-b_{j-1}^2{P}_{j-2}\left({\lambda }_j^{\left(j\right)}\right)-c_{j-2}^2{Q}_{j-1}\left({\lambda }_j^{\left(j\right)}\right)\hfill \\ +2b_{j-1}{c}_{j-2}{R}_{j-1}\left({\lambda }_j^{\left(j\right)}\right)=0.\hfill \end{array}\right.\end{array}$$

(11)

Solving (11), we obtain

$${\alpha }_j{X}^2+{\gamma }_j{Y}^2+2{\beta }_{j}XY+{\delta }_j=0.$$

(12)

with $X=b_{j-1}$ and $Y=c_{j-2}$. Note that, by fixing Y, the discriminant of Equation (12) is

$${▵}_X=4[({\beta }_j^2-{\alpha }_j{\gamma }_j)Y^2-{\alpha }_j{\delta }_j].$$

(13)

Since ${\alpha }_j{\delta }_j<0$, ${▵}_X>0$ if

(i)

${\beta }_j^2-{\alpha }_j{\gamma }_j>0$, for all $Y\in \mathbb{R}$.

(ii)

${\beta }_j^2-{\alpha }_j{\gamma }_j<0$, for $Y\in \left(-\sqrt{{\displaystyle \frac{{\alpha }_j{\delta }_j}{{\beta }_j^2-{\alpha }_j{\gamma }_j}}},\sqrt{{\displaystyle \frac{{\alpha }_j{\delta }_j}{{\beta }_j^2-{\alpha }_j{\gamma }_j}}}\right)$.

Thus, X exists in either case. Analogously, by fixing X, we obtain that Y also exists.

Moreover, the point $(X,Y)=(b_{j-1},c_{j-2})$ belongs to the conic

$$\mathcal{C}=\left\{(X,Y)\in {\mathbb{R}}^2:{\alpha }_j{X}^2+{\gamma }_j{Y}^2+2{\beta }_{j}XY+{\delta }_j=0\right\},$$

which, by Lemma 1 and condition (9), is non-degenerate, non-empty and centered at the origin. Therefore, there exist positive numbers $b_{j-1}$ and $c_{j-2}$, $j=3,4,\dots ,n$ satisfying (12).

Finally, from (11) and Lemma 1, the entries $a_j,j=3,4,\dots ,n$ are obtained. □

Theorem 2.

Let $2n-1$ be real numbers ${\left\{{\lambda }_1^{\left(j\right)},{\lambda }_j^{\left(j\right)}\right\}}_{j=1}^n$ and the vector $\mathbf{x}=(x_1,x_2,\dots ,x_n)$, satisfying

$${\lambda }_1^{\left(n\right)}<\cdots <{\lambda }_1^{\left(j\right)}<\cdots <{\lambda }_1^{\left(2\right)}<{\lambda }_1^{\left(1\right)}<{\lambda }_2^{\left(2\right)}<\cdots <{\lambda }_j^{\left(j\right)}<\cdots <{\lambda }_n^{\left(n\right)},$$

and

$$x_j{x}_{j+1}>0,i=1,2,\dots ,n-1.$$

(14)

If

$${\beta }_j^2-{\alpha }_j{\gamma }_j\ne 0$$

for $j=3,4,\dots n$, where ${\alpha }_j,{\beta }_j,{\gamma }_j$ and ${\delta }_j$ are as in (7), then there exists a unique symmetric pentadiagonal matrix $\mathcal{P}$ of the form (1), such that ${\lambda }_1^{\left(j\right)}$ and ${\lambda }_j^{\left(j\right)}$ are the extremal eigenvalues of the leading principal submatrix ${\mathcal{P}}_j$ of $\mathcal{P},j=1,2,\dots ,n$, and $({\lambda }_n^{\left(n\right)},\mathbf{x})$ is the eigenpair of $\mathcal{P}$.

Proof of Theorem 2.

From Theorem 1, by conditions (8) and (9), we have determined the existence of a pentadiagonal matrix with the required spectral properties.

Now, from System (10), for $j=1,2$, and Lemma 1, it is clear that

$$a_1={\lambda }_1^{\left(1\right)},$$

(15)

$$b_1=\sqrt{\left({\lambda }_1^{\left(1\right)}-{\lambda }_1^{\left(2\right)}\right)\left({\lambda }_2^{\left(2\right)}-{\lambda }_1^{\left(1\right)}\right)}$$

(16)

and

$$a_2={\lambda }_1^{\left(2\right)}+{\lambda }_2^{\left(2\right)}-{\lambda }_1^{\left(1\right)}.$$

(17)

Now, from the second equality in (10), it follows that

$$\left.\begin{array}{ccc}\hfill a_1{x}_1+b_1{x}_2+c_1{x}_3& =\hfill & \hfill {\lambda }_n^{\left(n\right)}{x}_1\\ \hfill b_1{x}_1+a_2{x}_2+b_2{x}_3+c_2{x}_4& =\hfill & \hfill {\lambda }_n^{\left(n\right)}{x}_2\\ \hfill c_{j-2}{x}_{j-2}+b_{j-1}{x}_{j-1}+a_j{x}_j+b_j{x}_{j+1}+c_j{x}_{j+2}& =\hfill & \hfill {\lambda }_n^{\left(n\right)}{x}_j,& \hfill & \hfill j=3,4,\dots ,n-2\\ \hfill c_{n-3}{x}_{n-3}+b_{n-2}{x}_{n-2}+a_{n-1}{x}_{n-1}+b_{n-1}{x}_n& =\hfill & \hfill {\lambda }_n^{\left(n\right)}{x}_{n-1}\\ \hfill c_{n-2}{x}_{n-2}+b_{n-1}{x}_{n-1}+a_n{x}_n& =\hfill & \hfill {\lambda }_n^{\left(n\right)}{x}_n.\end{array}\right\}$$

(18)

Thus, from (18) and condition (14), we obtain that

$$c_{j-2}=({\lambda }_n^{\left(n\right)}-a_{j-2}){\displaystyle \frac{x_{j-2}}{x_j}}-c_{j-4}\frac{x_{j-4}}{x_j}-b_{j-3}{\displaystyle \frac{x_{j-3}}{x_j}}-b_{j-2}\frac{x_{j-1}}{x_j}$$

(19)

for $j=3,4,\dots ,n$, where $c_{-1}=c_0=x_{-1}=x_0=b_0=0.$

On the other hand, from System (10), for $j=3,4,\dots ,n$, we have

$${\alpha }_j{b}_{j-1}^2+{\gamma }_j{c}_{j-2}^2+2{\beta }_j{b}_{j-1}{c}_{j-2}+{\delta }_j=0.$$

Then, from (19), Condition (9) and Lemma 1, we have

$$b_{j-1}=\frac{-c_{j-2}{\beta }_j\pm \sqrt{c_{j-2}^2{\beta }_j^2-{\alpha }_j(c_{j-2}^2{\gamma }_j+{\delta }_j)}}{{\alpha }_j}$$

(20)

and

$$a_j={\lambda }_i^{\left(j\right)}-{\displaystyle \frac{b_{j-1}^2{P}_{j-2}\left({\lambda }_i^{\left(j\right)}\right)+c_{j-2}^2{Q}_{j-1}\left({\lambda }_i^{\left(j\right)}\right)-2b_{j-1}{c}_{j-2}{R}_{j-1}\left({\lambda }_i^{\left(j\right)}\right)}{P_{j-1}\left({\lambda }_i^{\left(j\right)}\right)}}$$

(21)

for $j=3,4,\dots ,n$, and $i=1\vee j$. This concludes the proof. □

3. Construction of Nonsymmetric Pentadiagonal Matrix

In this section, we give sufficient conditions for the existence of a nonsymmetric full pentadiagonal matrix. It should be noted that the construction given in Theorem 3 and the similarity process of Lemma 3 provide a procedure to construct a nonsymmetric full pentadiagonal matrix. However, this has the disadvantage that only uniqueness is obtained in the diagonal entries, in contrast to the procedure presented below. An extension of Lemma 4 to the nonsymmetric case is given in the following lemma.

Lemma 5.

Let $\tilde{\mathcal{P}}$ be an $n\times n$ nonsymmetric pentadiagonal matrix of the form (2), and let ${\tilde{\mathcal{P}}}_j$ be the $j\times j$ principal submatrix of $\tilde{\mathcal{P}}$ with characteristic polynomial $P_j\left(\lambda \right)=det(\lambda I_j-{\tilde{\mathcal{P}}}_j),$$j=1,2,\dots ,n$. Then, the sequence ${\left\{P_j\left(\lambda \right)\right\}}_{j=1}^n$ satisfies the recurrence relation:

$$\begin{array}{ccc}\hfill P_1\left(\lambda \right)& =& \lambda -a_1,\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hfill \end{array}$$

(22)

$$\begin{array}{ccc}\hfill P_2\left(\lambda \right)& =& (\lambda -a_2)P_1\left(\lambda \right)-b_1{c}_1,\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hfill \end{array}$$

(23)

$$\begin{array}{ccc}\hfill P_j\left(\lambda \right)& =& (\lambda -a_j)P_{j-1}\left(\lambda \right)-b_{j-1}{c}_{j-1}{P}_{j-2}\left(\lambda \right)-d_{j-2}{e}_{j-2}{Q}_{j-1}\left(\lambda \right)\hfill \\ & & \hspace{1em}\hspace{1em}\hspace{1em}+b_{j-1}{e}_{j-2}{R}_{j-1}\left(\lambda \right)+c_{j-1}{d}_{j-2}{T}_{j-1}\left(\lambda \right);j=3,4,\dots ,n,\hfill \end{array}$$

(24)

where $Q_{j-1}\left(\lambda \right)$, $R_{j-1}\left(\lambda \right)$, and $T_{j-1}\left(\lambda \right)$ are the determinants of the submatrices, resulting from eliminating the $(j-2)$-th row and column, the $(j-1)$-th row and $(j-2)$-th column, and the $(j-2)$-th row and $(j-1)$-th column of submatrix $\lambda I_{j-1}-{\tilde{\mathcal{P}}}_{j-1}$, respectively.

Proof of Lemma 5.

It is immediate by expanding $det(\lambda I_j-{\tilde{\mathcal{P}}}_j)$. □

In the following, we consider the notations

$$\left.\begin{array}{cc}\hfill {\eta }_{j-1}& =b_{j-1}{P}_{j-2}\left({\lambda }_1^{\left(j\right)}\right)-d_{j-2}{T}_{j-1}\left({\lambda }_1^{\left(j\right)}\right)\hfill \\ \hfill {\theta }_{j-1}& =d_{j-2}{Q}_{j-1}\left({\lambda }_1^{\left(j\right)}\right)-b_{j-1}{R}_{j-1}\left({\lambda }_1^{\left(j\right)}\right)\hfill \\ \hfill {\mu }_{j-1}& =b_{j-1}{P}_{j-2}\left({\lambda }_j^{\left(j\right)}\right)-d_{j-2}{T}_{j-1}\left({\lambda }_j^{\left(j\right)}\right)\hfill \\ \hfill {\tau }_{j-1}& =d_{j-2}{Q}_{j-1}\left({\lambda }_j^{\left(j\right)}\right)-b_{j-1}{R}_{j-1}\left({\lambda }_j^{\left(j\right)}\right)\hfill \\ \hfill & \mathrm{for}j=3,4,\dots ,n,\hfill \end{array}\right\}$$

(25)

and

$$z_{h,k}=x_h{y}_k-x_k{y}_h\mathrm{for}h=j+1,j+2\mathrm{and}k=j-2,j-1,j.$$

(26)

Theorem 3.

Let $2n-1$ be real numbers ${\left\{{\lambda }_1^{\left(j\right)},{\lambda }_j^{\left(j\right)}\right\}}_{j=1}^n$, the vectors $\mathbf{x}=(x_1,x_2,\dots ,x_n)$ and $\mathbf{y}=(y_1,y_2,\dots ,y_{n-1})$, and real nonnegative numbers ${\ell }_1,{\ell }_2,\dots ,{\ell }_{n-2}$, satisfying

$${\lambda }_1^{\left(n\right)}<\cdots <{\lambda }_1^{\left(j\right)}<\cdots <{\lambda }_1^{\left(2\right)}<{\lambda }_1^{\left(1\right)}<{\lambda }_2^{\left(2\right)}<\cdots <{\lambda }_j^{\left(j\right)}<\cdots <{\lambda }_n^{\left(n\right)},$$

$$x_j{x}_{j+1}>0,j=1,2,\dots ,n-1,$$

$$y_j{y}_{j+1}>0,j=1,2,\dots ,n-2$$

(27)

and

$${\displaystyle \frac{x_{j+2}}{x_{j+1}}}\ne {\displaystyle \frac{y_{j+2}}{y_{j+1}}},j=1,2,\dots ,n-3.$$

(28)

If

$${\mu }_{j-1}{\theta }_{j-1-}{\tau }_{j-1}{\eta }_{j-1}\ne 0$$

(29)

for $j=3,4,\dots n$, where ${\eta }_j$, ${\theta }_j$, ${\mu }_j$ and ${\tau }_j$ are as in (25), then there exists a nonsymmetric pentadiagonal matrix $\tilde{\mathcal{P}}$ of the form (2), such that ${\lambda }_1^{\left(j\right)}$ and ${\lambda }_j^{\left(j\right)}$ are the extremal eigenvalues of the leading principal submatrix ${\tilde{\mathcal{P}}}_j$ of $\tilde{\mathcal{P}},j=1,2,\dots ,n$, $({\lambda }_n^{\left(n\right)},\mathbf{x})$ is the eigenpair of $\tilde{\mathcal{P}}$, $({\lambda }_{n-1}^{(n-1)},\mathbf{y})$ is the eigenpair of ${\tilde{\mathcal{P}}}_{n-1}$, and $a_{i+2}={\ell }_i;i=1,\dots ,n-2$.

Proof of Theorem 3.

Suppose that the set ${\left\{{\lambda }_1^{\left(j\right)},{\lambda }_j^{\left(j\right)}\right\}}_{j=1}^n$, and the vectors $\mathbf{x}$ and $\mathbf{y}$, satisfy (8) and (28), respectively. To show the existence of a symmetric pentadiagonal matrix $\tilde{\mathcal{P}}$ with the required properties is equivalent to showing that the system of equations

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}{P}_j\left({\lambda }_i^{\left(j\right)}\right)=0,\hfill & \mathrm{for}j=1,2,\dots ,n,\mathrm{and}i=1,j,\hfill \\ \tilde{\mathcal{P}}\mathbf{x}={\lambda }_n^{\left(n\right)}\mathbf{x},\hfill \\ \tilde{\mathcal{P}}\mathbf{y}={\lambda }_{n-1}^{\left(n-1\right)}\mathbf{y},\hfill \end{array}\right.\end{array}$$

(30)

where $P_j\left(\lambda \right)=det(\lambda I_j-{\tilde{\mathcal{P}}}_j),j=1,2,\dots ,n$ satisfies Lemma 5, has real solutions $a_j,b_j,c_j,d_j$ and $e_j$.

From the System (30) for $j=1$ and Condition (8), its is clear that $a_1={\lambda }_1^{\left(1\right)}$. Note that the second and third equalities in (30) have the form:

$$\left.\begin{array}{cc}\hfill a_1{x}_1+b_1{x}_2+d_1{x}_3& ={\lambda }_n^{\left(n\right)}{x}_1\hfill \\ \hfill c_1{x}_1+a_2{x}_2+b_2{x}_3+d_2{x}_4& ={\lambda }_n^{\left(n\right)}{x}_2\hfill \\ \hfill e_{j-2}{x}_{j-2}+c_{j-1}{x}_{j-1}+a_j{x}_j+b_j{x}_{j+1}+d_j{x}_{j+2}& ={\lambda }_n^{\left(n\right)}{x}_j,\hfill & \hfill j=3,4,\dots ,n-2\\ \hfill e_{n-3}{x}_{n-3}+c_{n-2}{x}_{n-2}+a_{n-1}{x}_{n-1}+b_{n-1}{x}_n& ={\lambda }_n^{\left(n\right)}{x}_{n-1}\hfill \\ \hfill e_{n-2}{x}_{n-2}+c_{n-1}{x}_{n-1}+a_n{x}_n& ={\lambda }_n^{\left(n\right)}{x}_n\hfill \\ \hfill a_1{y}_1+b_1{y}_2+d_1{y}_3& ={\lambda }_{n-1}^{(n-1)}{y}_1\hfill \\ \hfill c_1{y}_1+a_2{y}_2+b_2{y}_3+d_2{y}_4& ={\lambda }_{n-1}^{(n-1)}{y}_2\hfill \\ \hfill e_{j-2}{y}_{j-2}+c_{j-1}{y}_{j-2}+a_j{y}_j+b_j{y}_{j+1}+d_j{y}_{j+2}& ={\lambda }_{n-1}^{(n-1)}{y}_j,\hfill & \hfill j=3,4,\dots ,n-3\\ \hfill e_{n-4}{y}_{n-4}+c_{n-3}{y}_{n-3}+a_{n-2}{y}_{n-2}+b_{n-2}{y}_{n-1}& ={\lambda }_{n-1}^{(n-1)}{y}_{n-2}\hfill \\ \hfill e_{n-3}{y}_{n-3}+c_{n-2}{y}_{n-2}+a_{n-1}{y}_{n-1}& ={\lambda }_{n-1}^{(n-1)}{y}_{n-1}.\hfill \end{array}\right\}$$

(31)

Then, from (31) and Condition (28),

$$b_1={\displaystyle \frac{{\lambda }_n^{\left(n\right)}{x}_1{y}_2-{\lambda }_{n-1}^{(n-1)}{x}_2{y}_1+a_1(x_2{y}_1-x_1{y}_2)}{x_3{y}_2-x_2{y}_3}}$$

(32)

and

$$d_1={\displaystyle \frac{{\lambda }_n^{\left(n\right)}{x}_1{y}_3-{\lambda }_{n-1}^{(n-1)}{x}_3{y}_1+a_1(x_3{y}_1-x_1{y}_3)}{x_2{y}_3-x_3{y}_2}}.$$

(33)

From System (30) for $j=2$, Condition (8), and Lemma 1, we obtain

$$a_2={\displaystyle \frac{{\lambda }_1^{\left(2\right)}{P}_1\left({\lambda }_1^{\left(2\right)}\right)-{\lambda }_2^{\left(2\right)}{P}_1\left({\lambda }_2^{\left(2\right)}\right)}{P_1\left({\lambda }_1^{\left(2\right)}\right)-P_1\left({\lambda }_2^{\left(2\right)}\right)}}$$

(34)

and

$$c_1=\frac{1}{b_1}{\displaystyle \frac{\left({\lambda }_2^{\left(2\right)}-{\lambda }_1^{\left(2\right)}\right)P_1\left({\lambda }_1^{\left(2\right)}\right)P_1\left({\lambda }_2^{\left(2\right)}\right)}{P_1\left({\lambda }_1^{\left(2\right)}\right)-P_1\left({\lambda }_2^{\left(2\right)}\right)}}.$$

(35)

Now, from (31) and Condition (28), we have

$$b_j={\displaystyle \frac{{\lambda }_n^{\left(n\right)}{x}_j{y}_{j+2}-{\lambda }_{n-1}^{(n-1)}{x}_{j+2}{y}_j+a_j{z}_{j+2,j}+c_{j-1}{z}_{j+2,j-1}+e_{j-2}{z}_{j+2,j-2}}{x_{j+1}{y}_{j+2}-x_{j+2}-y_{j+1}}}$$

(36)

and

$$d_j={\displaystyle \frac{{\lambda }_n^{\left(n\right)}{x}_j{y}_{j+1}-{\lambda }_{n-1}^{(n-1)}{x}_{j+1}{y}_j+a_j{z}_{j+1,j}+c_{j-1}{z}_{j+1,j-1}+e_{j-2}{z}_{j+1,j-2}}{x_{j+2}{y}_{j+1}-x_{j+1}-y_{j+2}}}$$

(37)

for $j=2,3,\dots ,n-3$, where $e_0=x_0=y_0=0$.

On the other hand, from System (30) for $j=3,4,\dots ,n$, and Condition (29), it follows that

$$c_{j-1}={\displaystyle \frac{({\lambda }_j^{\left(j\right)}-a_j)P_{j-1}\left({\lambda }_j^{\left(j\right)}\right){\theta }_{j-1}-({\lambda }_1^{\left(j\right)}-a_j)P_{j-1}\left({\lambda }_1^{\left(j\right)}\right){\tau }_{j-1}}{{\mu }_{j-1}{\theta }_{j-1}-{\tau }_{j-1}{\eta }_{j-1}}}$$

(38)

and

$$e_{j-2}={\displaystyle \frac{({\lambda }_j^{\left(j\right)}-a_j)P_{j-1}\left({\lambda }_j^{\left(j\right)}\right){\eta }_{j-1}-({\lambda }_1^{\left(j\right)}-a_j)P_{j-1}\left({\lambda }_1^{\left(j\right)}\right){\mu }_{j-1}}{{\tau }_{j-1}{\eta }_{j-1}-{\mu }_{j-1}{\theta }_{j-1}}}.$$

(39)

Finally, from (31), and Conditions (14) and (27), we obtain

$$b_{n-2}=({\lambda }_{n-1}^{(n-1)}-a_{n-2}){\displaystyle \frac{y_{n-2}}{y_{n-1}}}-c_{n-3}\frac{y_{n-3}}{y_{n-1}}-e_{n-4}{\displaystyle \frac{y_{n-4}}{y_{n-1}}},$$

(40)

$$d_{n-2}=({\lambda }_n^{\left(n\right)}-a_{n-2}){\displaystyle \frac{x_{n-2}}{x_n}}-b_{n-2}\frac{x_{n-1}}{x_n}-c_{n-3}\frac{x_{n-3}}{x_n}-e_{n-4}{\displaystyle \frac{x_{n-4}}{x_n}}$$

(41)

and

$$b_{n-1}=({\lambda }_n^{\left(n\right)}-a_{n-1}){\displaystyle \frac{x_{n-1}}{x_n}}-c_{n-2}\frac{x_{n-2}}{x_n}-e_{n-3}{\displaystyle \frac{x_{n-3}}{x_n}}.$$

(42)

This completes the proof. □

4. Numerical Examples

The proofs of Theorems 2 and 3 provide algorithmic procedures for constructing symmetric and nonsymmetric pentadiagonal matrices from the information given in the respective statements. The Algorithms were implemented in Matlab R2023b. In this section, we present three examples showing the construction of these types of matrices satisfying the stated sufficiency conditions. To show the approximation of the spectral data obtained with respect to those initially, we consider the relative error

$$e_{\lambda }=log\left(\frac{{∥\lambda -\tilde{\lambda }∥}_2}{{∥\lambda ∥}_2}\right),e_{\mathbf{v}}=log\left(\frac{{∥\mathbf{v}-\tilde{\mathbf{v}}∥}_2}{{∥\mathbf{v}∥}_2}\right)\mathrm{and}{e}_{\mathbf{A}}=log\left(\frac{{∥\mathbf{A}-\tilde{\mathbf{A}}∥}_2}{{∥\mathbf{A}∥}_2}\right),$$

where $\lambda$ is the vector containing the information of the initially given eigenvalues and $\tilde{\lambda }$ of the eigenvalues of the constructed matrix; $\mathbf{v}$ represents the given eigenvector and $\tilde{\mathbf{v}}$ the eigenvector associated with a maximal eigenvalue of some leading principal submatrix of the constructed matrix; and $\mathbf{A}$ represents the initially given matrix and $\tilde{\mathbf{A}}$ the reconstructed matrix.

Example 1.

We use the spectral data given in

Table 1. Spectral data of matrix $\mathcal{P}$.

j	1	2	3	4	5	6	7
${\lambda }_1^{\left(j\right)}$	1.5062	1.4490	0.3876	0.3814	0.1965	0.1437	0.1265
${\lambda }_j^{\left(j\right)}$	1.5062	2.4118	2.7674	4.7154	4.9216	4.9317	4.9328
$x_j$	−0.0661	−0.3679	−0.1576	−0.8828	−0.2302	−0.0527	−0.0169

, which satisfies the conditions of Theorem 2.

Using the algorithmic procedure that follows from the proof of Theorem 2, we obtain the matrix

$$\mathcal{P}=\left(\begin{array}{ccccccc}\hfill 1.5052& \hfill 0.2277& \hfill 0.9049& & & & \\ \hfill 0.2277& \hfill 2.3546& \hfill 0.4357& \hfill 0.9797& & & \\ \hfill 0.9049& \hfill 0.4357& \hfill 1.1524& \hfill 0.3111& \hfill 0.4389& & \\ & \hfill 0.97978& \hfill 0.3111& \hfill 4.2215& \hfill 0.9234& \hfill 0.1111& \\ & & \hfill 0.4389& \hfill 0.9234& \hfill 0.9738& \hfill 0.4302& \hfill 0.2581\\ & & & \hfill 0.1111& \hfill 0.4302& \hfill 1.1296& \hfill 0.1848\\ & & & & \hfill 0.2581& \hfill 0.1848& \hfill 0.8535\end{array}\right),$$

whose spectral data, compared with the given data, have the relative errors shown in

Table 2. Relative errors in the construction of matrix $\mathcal{P}$.

${\mathit{e}}_{\mathit{\lambda }}$	${\mathit{e}}_{\mathit{x}}$
2.0164 × ${10}^{-16}$	1.5385 × ${10}^{-14}$

.

Example 2.

It is know that, for the matrix $J=tridiag\{1,2,1\}$,

$$J^2=\left(\begin{array}{ccccccc}\hfill 5& \hfill 4& \hfill 1& & & & \\ \hfill 4& \hfill 6& \hfill 4& \hfill 1& & & \\ \hfill 1& \hfill 4& \hfill 6& \hfill 4& \hfill 1& & \\ & \hfill \ddots & \hfill \ddots & \hfill \ddots & \hfill \ddots & \hfill \ddots & \\ & & \hfill 1& \hfill 4& \hfill 6& \hfill 4& \hfill 1\\ & & & \hfill 1& \hfill 4& \hfill 6& \hfill 4\\ & & & & \hfill 1& \hfill 4& \hfill 5\end{array}\right).$$

We reconstruct $J^2$ from its extremal eigenvalues and eigenpair $({\lambda }_n^{\left(n\right)},\mathbf{x})$ for different values of n. The results are shown in

Table 3. Relative errors in the reconstruction of matrix $J^2$.

n	${\mathit{e}}_{\mathit{\lambda }}$	${\mathit{e}}_{\tilde{\mathit{x}}}$	${\mathit{e}}_{{\mathit{J}}^2}$
5	$1.6018\times {10}^{-16}$	$1.7324\times {10}^{-15}$	$5.3559\times {10}^{-15}$
10	$1.9755\times {10}^{-16}$	$8.4638\times {10}^{-14}$	$8.2951\times {10}^{-13}$
15	$1.5504\times {10}^{-16}$	$8.7831\times {10}^{-12}$	$1.6018\times {10}^{-10}$
20	$2.0819\times {10}^{-16}$	$1.1785\times {10}^{-08}$	$3.2366\times {10}^{-07}$
25	$1.7147\times {10}^{-16}$	$1.1212\times {10}^{-04}$	$4.2308\times {10}^{-03}$

.

Example 3.

Considering the similarity process given in the proof of Lemma 3 and the condition (3), the matrix

$$\tilde{\mathcal{P}}=\left(\begin{array}{ccccccc}\hfill 8& \hfill 4& \hfill 64& & & & \\ \hfill 1& \hfill 9& \hfill 20& \hfill 32& & & \\ \hfill 4& \hfill 5& \hfill 11& \hfill 12& \hfill 5& & \\ & \hfill 2& \hfill 3& \hfill 12& \hfill 2& \hfill 3& \\ & & \hfill 5& \hfill 8& \hfill 13& \hfill 4& \hfill 1\\ & & & \hfill 48& \hfill 16& \hfill 15& \hfill 3\\ & & & & \hfill 64& \hfill 48& \hfill 16\end{array}\right)$$

is generated from matrix

$$\mathcal{P}=\left(\begin{array}{ccccccc}\hfill 8& \hfill 2& \hfill 16& & & & \\ \hfill 2& \hfill 9& \hfill 10& \hfill 8& & & \\ \hfill 16& \hfill 10& \hfill 11& \hfill 6& \hfill 5& & \\ & \hfill 8& \hfill 6& \hfill 12& \hfill 4& \hfill 12& \\ & & \hfill 5& \hfill 4& \hfill 13& \hfill 8& \hfill 8\\ & & & \hfill 12& \hfill 8& \hfill 15& \hfill 12\\ & & & & \hfill 8& \hfill 12& \hfill 16\end{array}\right).$$

We reconstruct these matrices from the spectral data required by Theorems 2 and 3.

Table 4. Spectral data of matrices $\mathcal{P}$ and $\tilde{\mathcal{P}}$.

j	1	2	3	4	5	6	7
${\lambda }_1^{\left(j\right)}$	8.0000	6.4384	−8.3931	−8.5420	−9.0139	−9.2627	−9.2636
${\lambda }_j^{\left(j\right)}$	8.0000	10.5616	29.4510	32.4643	33.6106	36.4775	39.8823
$x_j$	0.1874	0.2344	0.3441	0.4280	0.4055	0.5309	0.4026
${\tilde{x}}_j$	0.2110	0.1320	0.0969	0.0603	0.1142	0.2990	0.9070
${\tilde{y}}_j$	−0.7099	−0.3984	−0.2910	−0.1381	−0.1851	−0.4464

shows the given data, and

Table 5. Relative errors in the construction of matrix $\mathcal{P}$.

${\mathit{e}}_{\mathit{\lambda }}$	${\mathit{e}}_{\mathit{x}}$	${\mathit{e}}_{\mathcal{P}}$
$3.2171\times {10}^{-16}$	$6.0989\times {10}^{-14}$	$2.1021\times {10}^{-13}$

and

Table 6. Relative errors in the reconstruction of matrix $\tilde{\mathcal{P}}$.

${\mathit{e}}_{\mathit{\lambda }}$	${\mathit{e}}_{\tilde{\mathit{x}}}$	${\mathit{e}}_{\tilde{\mathit{y}}}$	${\mathit{e}}_{\tilde{\mathcal{P}}}$
$6.6467\times {10}^{-16}$	$2.7823\times {10}^{-14}$	$2.4069\times {10}^{-14}$	$5.8654\times {10}^{-13}$

show the results.

Example 4.

In the literature, the pentadiagonal symmetric Toeplitz matrices of the following form are well known:

$$\mathcal{T}=\left(\begin{array}{ccccccc}a& b& c& 0& 0& \dots & 0\\ b& a& b& c& 0& \dots & 0\\ c& b& a& \ddots & \ddots & \ddots & ⋮\\ 0& c& \ddots & \ddots & \ddots & \ddots & 0\\ 0& 0& \ddots & \ddots & \ddots & b& c\\ ⋮& ⋮& \ddots & \ddots & b& a& b\\ 0& 0& \dots & 0& c& b& a\end{array}\right).$$

(43)

In this example, we reconstruct these matrices where a, b, and c have arbitrary integer values between 1 and 20 and for different orders of the matrix. The results are shown in

Table 7. Relative errors in the reconstruction of matrix $\mathcal{T}$.

n	$\mathit{a},\mathit{b},\mathit{c}$	${\mathit{e}}_{\mathit{\lambda }}$	${\mathit{e}}_{\mathit{x}}$	${\mathit{e}}_{\mathcal{T}}$
5	$8,3,5$	$2.4139\times {10}^{-16}$	$2.6120\times {10}^{-15}$	$8.4554\times {10}^{-15}$
10	$2,1,4$	$3.3670\times {10}^{-16}$	$2.9225\times {10}^{-15}$	$3.8833\times {10}^{-14}$
15	$16,7,11$	$8.3783\times {10}^{-16}$	$7.8274\times {10}^{-15}$	$2.6996\times {10}^{-13}$
20	$14,14,15$	$5.6996\times {10}^{-16}$	$1.7073\times {10}^{-13}$	$1.6396\times {10}^{-11}$
25	$19,4,17$	$2.9743\times {10}^{-15}$	$1.2533\times {10}^{-13}$	$1.2488\times {10}^{-11}$
50	$9,3,20$	$5.0462\times {10}^{-15}$	$4.5855\times {10}^{-11}$	$9.6061\times {10}^{-9}$

.

5. Conclusions

This paper provides new procedures to reconstruct symmetric and nonsymmetric pentadiagonal matrices of order n from the minimal and maximal eigenvalues of all their leading principal submatrices. In the symmetric case, it is also necessary to consider an eigenvector of the maximal eigenvalue of the matrix. In the nonsymmetric case, an eigenvector of the maximal eigenvalue of the leading principal submatrix of order $n-1$ is additionally required. In both cases, we give sufficient conditions for the existence and realizability of such matrices. As our results are constructive, algorithmic procedures are generated to determine a solution matrix.