On the spectra of some \(g\)-circulant matrices and applications to nonnegative inverse eigenvalue problem (Q2174480)

A multi-set \(\Lambda = \{\lambda_1, \lambda_2, . . . , \lambda_n\}\) of complex numbers is said to be realizable if it is the spectrum, counting multiplicities, of a (entry-wise) nonnegative \(n\times n\) matrix \(A\). The problem of finding necessary and sufficient conditions for \(\Lambda\) being realizable is known as the nonnegative inverse eigenvalue problem (NIEP). A \(g\)-circulant matrix is an \(n\times n\) matrix \(A\) whose elements of each row of \(A\) are identical to those of the previous row, but are shifted \(g\) positions to the right and wrapped around. So the circulant matrix is the special case when \(g=1\). See [\textit{P. J. Davis}, Circulant matrices. 2nd ed. New York, NY: AMS Chelsea Publishing (1994; Zbl 0898.15021)] for the classical theory of circulant matrix. In this paper, the authors characterize the spectra of some real \(g\)-circulant matrices and the results are applied to NIEP to construct nonnegative, \(g\)-circulant matrices with prescribed spectrum. They also show that for certain positive integers \(n\) and \(g\), a \(g\)-circulant matrix is completely determined by its diagonal entries. Many results including their proofs have a strong flavor of number theory. The presence of many examples and problems is a special feature of the paper.