## 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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