Abstract
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.
Highlights
Circulant and skew-circulant matrices are appearing increasingly often in scientific and engineering applications
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
On the basis of existing application situation [4], we conjecture that SVD decomposition of skew circulant matrix will play an important role in CT-perfusion imaging of human brain
Summary
Circulant and skew-circulant matrices are appearing increasingly often in scientific and engineering applications. Shen et al considered circulant matrices with the Fibonacci and Lucas numbers and presented their explicit determinants and inverses by constructing the transformation matrices [22]. Gao et al [23] gave explicit determinants and inverses of skew circulant and skew left circulant matrices with the Fibonacci and Lucas numbers. Journal of Applied Mathematics and skew left circulant matrices with the k-Fibonacci numbers and the k-Lucas numbers and discussed the invertibility of the these matrices and presented their determinant and the inverse matrix by constructing the transformation matrices, respectively. Solak [26] established the lower and upper bounds for the spectral norms of circulant matrices with the classical Fibonacci and Lucas numbers entries. 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.
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