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I am using a square matrix that has the same values on the diagonal (say, alpha) and a different value for all diagonal elements (say, beta). Does this type of matrix have a name? It is such a special form of a Toeplitz matrix that it surely must have its own name!

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Are you describing a Circulant matrix? –  Mufasa Sep 11 '13 at 21:29
There is quite an extensive list of matrices here –  Mufasa Sep 11 '13 at 21:39
It is also a special case of a circulant matrix. The a 4d version of the matrix is {a, b, b, b//b, a, b, b//b, b, a, b//b, b, b, a}. Thank you for the list... I will look through it thoroughly! –  cma Sep 11 '13 at 22:05

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