Circulant Type Matrices with the Sum and Product of Fibonacci and Lucas Numbers

Abstract

Circulant type matrices have become an important tool in solving differential equations. In this paper, we consider circulant type matrices, including the circulant and left circulant and<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M1"><mml:mrow><mml:mi>g</mml:mi></mml:mrow></mml:math>-circulant matrices with the sum and product of Fibonacci and Lucas numbers. Firstly, we discuss the invertibility of the circulant matrix and present the determinant and the inverse matrix by constructing the transformation matrices. Furthermore, the invertibility of the left circulant and<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M2"><mml:mrow><mml:mi>g</mml:mi></mml:mrow></mml:math>-circulant matrices is also discussed. We obtain the determinants and the inverse matrices of the left circulant and<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M3"><mml:mrow><mml:mi>g</mml:mi></mml:mrow></mml:math>-circulant matrices by utilizing the relation between left circulant, and<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M4"><mml:mrow><mml:mi>g</mml:mi></mml:mrow></mml:math>-circulant matrices and circulant matrix, respectively.