Is there necessary and sufficient condition on whether a symmetric circulant matrix is non-singular? I found many example supporting that positive symmetric circulant matrices which has at least two different entries are non-singular.

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    $\begingroup$ Clearly$$\pmatrix{a&b&a&b\cr b&a&b&a\cr a&b&a&b\cr b&a&b&a\cr}$$is singular. $\endgroup$ – David Dec 15 '14 at 5:45

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