Some identities of the generalized bi-periodic Fibonacci and Lucas polynomials

1.   Introduction

Since the 1960s, the rapid development of high-speed rail has made it a very important means of transportation. However, the vibration will be caused because of the contact between the wheels of the train and the train tracks during the operation of the high-speed train. Therefore, the analytical vibration model can be mathematically summarized as a quadratic palindromic eigenvalue problem (QPEP) (see ^[1,2])

with $A_i \in {\mathbb{R}}^{n \times n}, \ i = 0, 1$ and $A_0^\top = A_0.$ The eigenvalues $\lambda$, the corresponding eigenvectors $x$ are relevant to the vibration frequencies and the shapes of the vibration, respectively. Many scholars have put forward many effective methods to solve QPEP ^[3,4,5,6,7]. In addition, under mild assumptions, the quadratic palindromic eigenvalue problem can be converted to the following linear palindromic eigenvalue problem (see ^[8])

with $A \in {\mathbb{R}}^{n \times n}$ is a given matrix, $\lambda\in {\mathbb{C}}$ and nonzero vectors $x\in {\mathbb{C}}^{n}$ are the wanted eigenvalues and eigenvectors of the vibration model. We can obtain $\frac{1}{\lambda} x^\top A^\top = x^\top A$ by transposing the equation (1). Thus, $\lambda$ and $\frac{1}{\lambda}$ always come in pairs. Many methods have been proposed to solve the palindromic eigenvalue problem such as URV-decomposition based structured method ^[9], QR-like algorithm ^[10], structure-preserving methods ^[11], and palindromic doubling algorithm ^[12].

On the other hand, the modal data obtained by the mathematical model are often evidently different from the relevant experimental ones because of the complexity of the structure and inevitable factors of the actual model. Therefore, the coefficient matrices need to be modified so that the updated model satisfies the dynamic equation and closely matches the experimental data. Al-Ammari ^[13] considered the inverse quadratic palindromic eigenvalue problem. Batzke and Mehl ^[14] studied the inverse eigenvalue problem for T-palindromic matrix polynomials excluding the case that both $+1$ and $-1$ are eigenvalues. Zhao et al. ^[15] updated $\ast$-palindromic quadratic systems with no spill-over. However, the linear inverse palindromic eigenvalue problem has not been extensively considered in recent years.

In this work, we just consider the linear inverse palindromic eigenvalue problem (IPEP). It can be stated as the following problem:

Problem IPEP. Given a pair of matrices $(\Lambda, X)$ in the form

and

where diagonal elements of $\Lambda$ are all distinct, $X$ is of full column rank $p$, and both $\Lambda$ and $X$ are closed under complex conjugation in the sense that $\lambda_{2i} = \bar{\lambda}_{2i-1} \in {\mathbb{C}}$, $x_{2i} = \bar{x}_{2i-1} \in {\mathbb{C}}^{n}$ for $i = 1, \cdots, m$, and $\lambda_{j} \in {\mathbb{R}}$, $x_{j} \in {\mathbb{R}}^{n}$ for $j = 2m+1, \cdots, p$, find a real-valued matrix $A$ that satisfy the equation

Namely, each pair $(\lambda_{t}, x_{t})$, $t = 1, \cdots, p$, is an eigenpair of the matrix pencil

It is known that the mathematical model is a "good" representation of the system, we hope to find a model that is closest to the original model. Therefore, we consider the following best approximation problem:

Problem BAP. Given $\tilde{A} \in {\mathbb{R}}^{n \times n}$, find $\hat{A} \in {\mathcal{S}}_{A}$ such that

where $\| \cdot \|$ is the Frobenius norm, and ${\mathcal{S}}_{A}$ is the solution set of Problem IPEP.

In this paper, we will put forward a new direct method to solve Problem IPEP and Problem BAP. By partitioning the matrix $\Lambda$ and using the QR-decomposition, the expression of the general solution of Problem IPEP is derived. Also, we show that the best approximation solution $\hat{A}$ of Problem BAP is unique and derive an explicit formula for it.

2.   The solution of Problem IPEP

We first rearrange the matrix $\Lambda$ as

where $t+2s+2(k+2l) = p$, $t = 0 \ or \ 1,$

with

and the adjustment of the column vectors of $X$ corresponds to those of $\Lambda$.

Define $T_p$ as

where $i = \sqrt{-1}$. It is easy to verify that $T_p^\mathrm{H} T_p = I_p$. Using this matrix of (5), we obtain

where

and $\tilde{\Lambda}_2 \in {\bf \mathbb{R}}^{2(k+2l) \times 2(k+2l)}$, $\tilde{X} \in {\bf \mathbb{R}}^{n \times p}$. $y_{t+2s+j}$ and $z_{t+2s+j}$ are, respectively, the real part and imaginary part of the complex vector $x_{t+2s+j}$ for $j = 1, 3, \cdots, 2(k+2l)-1$. Using (6) and (7), the matrix equation (2) is equivalent to

Since $\text{rank}(X) = \text{rank}(\tilde{X}) = p$. Now, let the QR-decomposition of $\tilde{X}$ be

where $Q = [Q_1, Q_2]\in \mathbb{R}^{n \times n}$ is an orthogonal matrix and $R \in \mathbb{R}^{p \times p}$ is nonsingular. Let

Using (9) and (10), then the equation of (8) is equivalent to

Write

then the equation of (11) is equivalent to

Because the elements of $\Lambda_1, \tilde{\Lambda}_2$ are distinct, we can obtain the following relations by Eqs (14)-(18)

where

and $1\leq i\leq k, 1\leq j\leq l$. $h_{1}, \cdots, h_{s}, a_{1}, \cdots, a_{k+2l}$ are arbitrary real numbers. It follows from Eq (12) that

where $E = R\tilde{\Lambda}R^{-1}$.

Theorem 1. Suppose that $\Lambda = \mathit{\text{diag}} \{\lambda_{1}, \cdots, \lambda_{p}\} \in {\mathbb{C}}^{p \times p}$, $X = [x_{1}, \cdots, x_{p}] \in {\mathbb{C}}^{n \times p}$, where diagonal elements of $\Lambda$ are all distinct, $X$ is of full column rank $p$, and both $\Lambda$ and $X$ are closed under complex conjugation in the sense that $\lambda_{2i} = \bar{\lambda}_{2i-1} \in {\mathbb{C}}$, $x_{2i} = \bar{x}_{2i-1} \in {\mathbb{C}}^{n}$ for $i = 1, \cdots, m$, and $\lambda_{j} \in {\mathbb{R}}$, $x_{j} \in {\mathbb{R}}^{n}$ for $j = 2m+1, \cdots, p$. Rearrange the matrix $\Lambda$ as $(4)$, and adjust the column vectors of $X$ with corresponding to those of $\Lambda$. Let $\Lambda, X$ transform into $\tilde{\Lambda}, \tilde{X}$ by $(6)-(7)$ and QR-decomposition of the matrix $\tilde{X}$ be given by $(9)$. Then the general solution of $(2)$ can be expressed as

where $E = R\tilde{\Lambda}R^{-1},$ $f_{11}$ is arbitrary real number, $A_{12} \in {\mathbb{R}}^{p \times (n-p)}, A_{22}\in {\mathbb{R}}^{(n-p) \times (n-p)}$ are arbitrary real-valued matrices and $F_{22}, F_{33}$ are given by $(20)-(21)$.

3.   The solution of Problem BAP

In order to solve Problem BAP, we need the following lemma.

Lemma 1. ^[16] Let $A, B$ be two real matrices, and $X$ be an unknown variable matrix. Then

By Theorem $1$, we can obtain the explicit representation of the solution set ${\mathcal{S}}_{A}$. It is easy to verify that $\mathcal{S}_A$ is a closed convex subset of ${\bf \mathbb{R}}^{n \times n}\times {\bf \mathbb{R}}^{n \times n}.$ By the best approximation theorem (see Ref. ^[17]), we know that there exists a unique solution of Problem BAP. In the following we will seek the unique solution $\hat{A}$ in $\mathcal{S}_A.$ For the given matrix $\tilde{A} \in {\bf \mathbb{R}}^{n \times n},$ write

then

Therefore, $\|A-\tilde{A}\| = \min$ if and only if

Let

then the relation of (25) is equivalent to

Write

where $r_{1, t} \in {\mathbb{R}}^{t \times p}, r_{2, i} \in {\mathbb{R}}^{1 \times p}, r_{3, j} \in {\mathbb{R}}^{2 \times p}, \ i = 1, \cdots, 2s, \ j = 1, \cdots, k+2l$.

Let

with $r = t+s, q = t+s+k+2l.$ Then the relation of (29) is equivalent to

that is,

where $c_{i, j} = \text{tr} (J_i^\top J_j), e_i = \text{tr} (J_i^\top\tilde{A}_{11}) (i, j = t, \cdots, t+s+k+2l)$ and $c_{i, j} = c_{j, i}$.

Consequently,

Clearly, $g(f_{11}, h_1, \cdots, h_s, a_1, \cdots, a_{k+2l}) = \text{min}$ if and only if

Therefore,

If let

where $C$ is symmetric matrix. Then the equation (32) is equivalent to

and the solution of the equation (33) is

Substituting (34) into (20)-(21), we can obtain $f_{11}, F_{22}$ and $F_{33}$ explicitly. Similarly, the equation of (26) is equivalent to

Applying Lemma 1, we obtain

setting $\frac{\partial g(A_{12})}{\partial A_{12}} = 0$, we obtain

Theorem 2. Given $\tilde{A} \in {\mathbb{R}}^{n \times n}$, then the Problem BAP has a unique solution and the unique solution of Problem BAP is

where $E = R\tilde{\Lambda} R^{-1}$, $F_{22}, F_{33}, A_{12}, \tilde{A}_{22}$ are given by $(20), (21), (35), (24)$ and $f_{11}, h_{1}, \cdots, h_s, a_1, \cdots, a_{k+2l}$ are given by $(34)$.

4.   A numerical example

Based on Theorems $1$ and $2$, we can describe an algorithm for solving Problem BAP as follows.

Algorithm 1.

1) Input matrices $\Lambda$, $X$ and $\tilde{A}$;

2) Rearrange $\Lambda$ as (4), and adjust the column vectors of $X$ with corresponding to those of $\Lambda$;

3) Form the unitary transformation matrix $T_p$ by (5);

4) Compute real-valued matrices $\tilde{\Lambda}, \tilde{X}$ by (6) and (7);

5) Compute the QR-decomposition of $\tilde{X}$ by (9);

6) $F_{12} = 0, F_{21} = 0, F_{13} = 0, F_{31} = 0, F_{23} = 0, F_{32} = 0$ by (19) and $E = R\tilde{\Lambda} R^{-1}$;

7) Compute $\tilde{A}_{ij} = Q_i^\top \tilde{A}Q_j, i, j = 1, 2$;

8) Compute $R^{-1}$ by (28) to form $R_1, R_2, R_3$;

9) Divide matrices $R_1, R_2, R_3$ by (30) to form $r_{1, t}, r_{2, i}, r_{3, j},$ $i = 1, \cdots, 2s, j = 1, \cdots, k+2l$;

10) Compute $J_i,$ $i = t, \cdots, t+s+k+2l,$ by (31);

11) Compute $c_{i, j} = \mbox{tr} (J_i^\top J_j), e_i = \mbox{tr} (J_i^\top\tilde{A}_{11}),$ $i, j = t, \cdots, t+s+k+2l$;

12) Compute $f_{11}, h_1, \cdots, h_s, a_1, \cdots, a_{k+2l}$ by (34);

13) Compute $F_{22}, F_{33}$ by (20), (21) and $A_{22} = \tilde{A}_{22}$;

14) Compute $A_{12}$ by (35) and $A_{21}$ by (22);

15) Compute the matrix $\hat{A}$ by (36).

Example 1. Consider a $11$-DOF system, where

the measured eigenvalue and eigenvector matrices $\Lambda$ and $X$ are given by

and

Using Algorithm 1, we obtain the unique solution of Problem BAP as follows:

and

Therefore, the new model $\hat{A}X = \hat{A}^\top X\Lambda$ reproduces the prescribed eigenvalues (the diagonal elements of the matrix $\Lambda$) and eigenvectors (the column vectors of the matrix $X$).

Example 2. (Example 4.1 of ^[12]) Given $\alpha = \cos(\theta)$, $\beta = \sin(\theta)$ with $\theta = 0.62$ and $\lambda_1 = 0.2, \lambda_2 = 0.3, \lambda_3 = 0.4$. Let

where $\Gamma = \left[{\begin{array}{cc} \alpha & -\beta\\ \beta & \alpha\\ \end{array}} \right].$ We construct

the measured eigenvalue and eigenvector matrices $\Lambda$ and $X$ are given by

and

Using Algorithm 1, we obtain the unique solution of Problem BAP as follows:

and

Therefore, the new model $\hat{A}X = \hat{A}^\top X\Lambda$ reproduces the prescribed eigenvalues (the diagonal elements of the matrix $\Lambda$) and eigenvectors (the column vectors of the matrix $X$).

5.   Concluding remarks

In this paper, we have developed a direct method to solve the linear inverse palindromic eigenvalue problem by partitioning the matrix $\Lambda$ and using the QR-decomposition. The explicit best approximation solution is given. The numerical examples show that the proposed method is straightforward and easy to implement.

Conflict of interest

The authors declare no conflict of interest.