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Drazin inverse of singular adjacency matrices of directed weighted cycles. (English) Zbl 1469.15007

Let \(C=C_n(a,b)\) denote the circulant matrix whose first row is given by the vector \((0,a,0,\ldots, 0,b)^T \in \mathbb{R}^n\), for a positive integer \(n\). First, the authors show that \(C\) is singular if and only if, either \(b=-a\) (independent of what \(n\) is), or \(b=a\) together with the condition that \(n\) is a multiple of four. These circulant matrices arise as adjacency matrices of directed weighted cycles, providing the motivation for their consideration. In this interesting work, the authors present explicit formulae for the Drazin inverses in each of the cases above.

MSC:

15A09 Theory of matrix inversion and generalized inverses
15B05 Toeplitz, Cauchy, and related matrices
05C50 Graphs and linear algebra (matrices, eigenvalues, etc.)
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