Circulant and skew circulant matrices have become an important tool in networks engineering. In this paper, we consider skew circulant type matrices with any continuous Fibonacci numbers. We discuss the invertibility of the skew circulant type matrices and present explicit determinants and inverse matrices of them by constructing the transformation matrices. Furthermore, the maximum column sum matrix norm, the spectral norm, the Euclidean (or Frobenius) norm, and the maximum row sum matrix norm and bounds for the spread of these matrices are given, respectively.

Skew circulant and circulant matrices have important applications in various networks engineering. Joy and Tavsanoglu [

The skew circulant matrices as preconditioners for linear multistep formulae- (LMF-) based ordinary differential equations (ODEs) codes, Hermitian, and skew-Hermitian Toeplitz systems were considered in [

Besides, some scholars have given various algorithms for the determinants and inverses of nonsingular circulant matrices. Unfortunately, the computational complexity of these algorithms is very amazing huge with the order of matrix increasing. However, some authors gave the explicit determinants and inverses of circulant and skew circulant matrices involving some famous numbers. For example, Yao and Jiang [

Recently, there are several papers on the norms of some special matrices. Solak [

Beginning with Mirsky [

The Fibonacci sequences are defined by the following recurrence relations [

The

The Fibonacci sequences were introduced for the first time by the famous Italian mathematician Leonardo of Pisa (nicknamed Fibonacci). It is well known that the ratio of two consecutive classical Fibonacci numbers converges to the golden mean, or the golden section,

The purpose of this paper is to obtain the explicit determinants, explicit inverses, norm, and spread of skew circulant type matrices involving any continuous Fibonacci numbers. And we generalize the result [

In the following, let

A skew circulant matrix with the first row

A skew left circulant matrix with the first row

Let

In this section, let

Let

Obviously,

Let

Taking

Let the matrix

Let

Then

Hence, we get

Let

Let

Since the last row elements of the matrix

Let

By Definition 8 in [

According to Definition 8 in [

Let

By Lemma 3 in [

Since

Since all skew circulant matrices are normal, by Lemma 9 in [

Let

The trace of

In this section, let

According to Lemma 5 in [

Let

Let

Let

Let

Using the method in Theorem

Let

According to Lemma 4 in [

Let

Since

By (18) in [

We discuss the invertibility of skew circulant type matrices with any continuous Fibonacci numbers and present the determinant and the inverse matrices by constructing the transformation matrices. The four kinds of norms and bounds for the spread of these matrices are given, respectively. 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 and Technology of Shandong Province (Grant no. 2012GGX10115) and NSFC (Grant no. 11301251) and the AMEP of Linyi University, China.

_{∞}high-energy physics