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On certain generalized circulant matrices. (English) Zbl 1052.15019

Summary: Let \(h\), \(n\) be positive integers, where \(1\leq h<n\), \(k=(n,h)\) and \(n=kn'\). We call \(h\)-generalized circulant a matrix \(A\) of order \(n\) which can be partitioned into \(h\)-circulant submatrices of type \(n'\times n\). We determine a characterization of \(h\)-generalized circulant matrices and, using this result, we prove that \(A=\sum^{\lfloor\frac{n}{h}\rfloor}_{j=0}a_jP^{jh}_n\) is permutation similar to the direct sum of \(k\) matrices coinciding with \(\sum^{\lfloor\frac{n}{h}\rfloor}_{j=1}a_jP^j_{n'}\), where \(P_n\) denote the \((0,1)\)-circulant matrix of order \(n\) whose first row is null but the element in position \((1, 2)\). This implies new results on the values of the permanent and also on the determination of the eigenvalues of \((0,1)\)-circulant matrices. A partial proof of a conjecture on the maximum value of permanents is achieved.

MSC:

15B57 Hermitian, skew-Hermitian, and related matrices
15A18 Eigenvalues, singular values, and eigenvectors
15A15 Determinants, permanents, traces, other special matrix functions
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