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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

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 [

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 [

Furthermore, circulant type matrices have been put on the firm basis with the work in [

The VanderLaan sequences are defined by the following recurrence relation [

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

The characteristic equation of VanderLaan numbers is

Some authors have presented the explicit determinants and inverse of circulant type matrices involving famous numbers in recent years. For example, in [

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

A VanderLaan left circulant matrix is an

A VanderLaan

Define the

A calculation using the expansion of the last column for determinant of matrix

We have

Let

Expanding the last column for the determinant of matrix

Let

From direct calculation by matrix multiplication, we get

Let the matrix

Let

We first introduce the following notations:

In this segment, let

Let

Apparently,

Let

According to Theorem

Let

Let

By Lemma

In this section, we gain three kinds of norms and lower bound for the spread of VanderLaan circulant matrix separately.

Let

By definition of norms in [

Let

According to Theorem 2 in [

Let

By (19) in [

In this part, let

According to Lemma 2 in [

Let

Let

Let

Let

By definition of norms in [

Let

From Theorem 2 in [

Let

By (19) in [

In this section, let

From Lemmas 3 and 4 in [

Let

Let

Let

Let

According to the definition of norms, Lemmas 3 and 4 in [

This completes the proofs.

In this paper, we considered VanderLaan circulant type matrices. We discussed the nonsingularity of these special matrices and presented the exact determinants and inverse matrices of VanderLaan circulant type 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

Furthermore, based on circulant matrices technology we will consider solving the problem in [

The authors declare that there is no conflict of interests regarding the publication of this paper.

The research was supported by the Development Project of Science & Technology of Shandong Province (Grant no. 2012GGX10115) and the AMEP of Linyi University, China.