Abstract

Circulant matrices have become a satisfactory tools in control methods for modern complex systems. In the paper, VanderLaan circulant type matrices are presented, which include VanderLaan circulant, left circulant, and -circulant matrices. The nonsingularity of these special matrices is discussed by the surprising properties of VanderLaan numbers. The exact determinants of VanderLaan circulant type matrices are given by structuring transformation matrices, determinants of well-known tridiagonal matrices, and tridiagonal-like matrices. The explicit inverse matrices of these special matrices are obtained by structuring transformation matrices, inverses of known tridiagonal matrices, and quasi-tridiagonal matrices. Three kinds of norms and lower bound for the spread of VanderLaan circulant and left circulant matrix are given separately. And we gain the spectral norm of VanderLaan -circulant matrix.

1. Introduction

It is well known that the circulant matrices are one of the most important research tools in control methods for modern complex systems. There are many results on circulant systems. The concept of “chart” was used to investigate structural controllability and fixed models in [1] while more attention was paid to properties and controller design for a special class of circulant systems, called symmetrically circulant systems [2–5].

Systems with block symmetric circulant structure are a kind of complex systems and constitute an important class of large-scale systems where there may be a clear advantage of using multivariable control and where it is actually used in practice. Typical application examples can be found in the control of paper machine [6–8] and control of power systems with parallel structure [7]. Other examples, including multizone crystal growth furnaces and dyes for plastic films, were also listed in [7, 8] and the references therein. For those symmetric circulant composite systems, due to the high dimensionality of the overall system and information structural constraints, the control problem is more complex and few results on regional pole assignment are available in the literature. In [9], the authors considered the problem of placing the poles of uncertain symmetric circulant composite systems in a specified disk, which is also presented as the problem of quadratic D stabilisation. Lee et al. [10] presented linear quadratic (LQ) repetitive control (RC) methods for processes represented by a conventional FIR model and a circulant FIR model. The latter, which represents a FIR system under the assumption of a cyclic steady state, is named its input-output map. The map is represented by a circulant matrix. Using the complete frequency resolving property of a circulant matrix, a special tuning method for the LQ weights is proposed. Lee and Won considered properties of pulse response circulant matrix and applied that to MIMO control and identification in [11].

Furthermore, circulant type matrices have been put on the firm basis with the work in [12–18] and so on. These special matrices have significant applications in various disciplines.

The VanderLaan sequences are defined by the following recurrence relation [19]: where

For the convenience of readers, we gave the first few values of the sequences as follows:

The characteristic equation of VanderLaan numbers is , and its roots are denoted by , , ; then The Binet form for VanderLaan sequence is where

Some authors have presented the explicit determinants and inverse of circulant type matrices involving famous numbers in recent years. For example, in [14], the nonsingularity of circulant type matrices with the sum and product of Fibonacci and Lucas numbers is discussed. And the exact determinants and inverses of these matrices are given. Determinants and inverses of circulant matrices with Jacobsthal and Jacobsthal-Lucas numbers are given in [20]. Jiang et al. [15] studied circulant type matrices with the -Fibonacci and -Lucas numbers and obtained the explicit determinants and inverse matrices. Lin [21] proposed the determinants of the Fibonacci-Lucas quasi-cyclic matrices. In [22], Jiang et al. discussed the nonsingularity of the skew circulant type matrices and presented explicit determinants and inverse matrices of these special matrices. Furthermore, four kinds of norms and bounds for the spread of these matrices are given separately. In [23], Shen et al. discussed circulant matrices with Fibonacci and Lucas numbers and gave their explicit determinants and inverses. Authors [24] discussed the nonsingularity of the circulant matrix and presented the explicit determinant and inverse matrices. Moreover, the nonsingularity of the left circulant and -circulant matrices is also studied. The explicit determinants and inverse matrices of the left circulant and -circulant matrices are obtained by utilizing the relationship between left circulant and -circulant matrices and circulant matrix, respectively.

This paper is aimed at getting the more beautiful results for the determinants and inverses of circulant type matrices via some surprising properties of VanderLaan numbers.

A VanderLaan circulant matrix is an matrix of the following form:

A VanderLaan left circulant matrix is an matrix of the following form:

A VanderLaan -circulant matrix is an matrix with the following form: where is a nonnegative integer and each of the subscripts is understood to be reduced modulo .

Lemma 1. Define the matrix by then determinant of the matrix is

Proof. A calculation using the expansion of the last column for determinant of matrix shows that ; let , ; then let , be the roots of the equation .
We have where We obtain

Lemma 2. Let be a  matrix; then where, for , and .

Proof. Expanding the last column for the determinant of matrix and according to Lemma 1, we obtain This completes the proof.

Lemma 3. Let be a partitioned matrix; then where , is a row vector, and is a column vector.

Proof. From direct calculation by matrix multiplication, we get where

Lemma 4. Let the matrix be of the form then the inverse of the matrix is equal to where

Proof. Let ; if , we have ; distinctly, for . And, for , we have We obtain , where is identity matrix. Evidenced by the same token, . The proof is completed.

We first introduce the following notations:

2. Determinant and Inverse of VanderLaan Circulant Matrix

In this segment, let be a VanderLaan circulant matrix. To begin with, we give the determinant of the matrix . And then we draw a conclusion that is an invertible matrix for ; finally we find the inverse of the matrix .

Theorem 5. Let be a VanderLaan circulant matrix, so its determinant is where is the th VanderLaan number.

Proof. Apparently, meets formula (27). If , let be two matrices; we have where we obtain Let be two matrices, and By Lemmas 1 and 2, the following equations hold: while we have which completes the proof.

Theorem 6. Let be a VanderLaan circulant matrix; in the case of , is a nonsingular matrix.

Proof. According to Theorem 5, in case of , we have . When , since , where we get where If there exists   showing that , here we get for . If , thus, is a real number. While , we obtain for . But is not the root of the equation . We have for any , while . If , we get that is an imaginary number, if and only if We obtain , such that . If , we have , , or ; obviously, and . In the same way, we know that and . So, , . By Lemma 1 in [14], the proof is obtained.