We consider the skew circulant and skew left circulant matrices with any continuous Lucas numbers. Firstly, we discuss the invertibility of the skew circulant matrices and present the determinant and the inverse matrices by constructing the transformation matrices. Furthermore, the invertibility of the skew left circulant matrices is also discussed. We obtain the determinants and the inverse matrices of the skew left circulant matrices by utilizing the relationship between skew left circulant matrices and skew circulant matrix, respectively. Finally, the four kinds of norms and bounds for the spread of these matrices are given, respectively.

Circulant and skew-circulant matrices are appearing increasingly often in scientific and engineering applications. Briefly, scanning the recent literature, one can see their utility is appreciated in the design of digital filters [

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

Besides, some scholars have given various algorithms for the determinants and inverses of nonsingular circulant matrices [

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

Beginning with Mirsky [

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

In the following, let

The Lucas sequences are defined by the following recurrence relations [

The

A skew circulant matrix over

A skew left circulant matrix over

Let

where

if

Let

With the orthogonal skew left circulant matrix

If

Let

Let

If

Let

Beginning with Mirsky [

Let

Let

if

and if

In this section, let

In the following, let

Let

Obviously,

So it holds that

Let

Taking

Let the matrix

Let

Let

Let

Then, we have

Since the last row elements of the matrix

Let

By Definition

Let

By Lemma

Since

Since all skew circulant matrices are normal, by Lemma

Let

The trace of

In this section, let

According to Lemmas

Let

Let

Let

Let

Using the method in Theorem

Let

According to Lemma

Let

Since

We discuss the invertibility of the skew circulant type matrices with any continuous Lucas 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.

This research was supported by the Natural Science Foundation of Shandong Province (Grant no. ZR2011FL017), the National Nature Science Foundation of China (Grant no. F020701), and the AMEP of Linyi University, China.