Noncirculant Toeplitz Matrices All of Whose Powers Are Toeplitz

Let a, b and c be fixed complex numbers. Let Mn(a, b, c) be the n×n Toeplitz matrix all of whose entries above the diagonal are a, all of whose entries below the diagonal are b, and all of whose entries on the diagonal are c. For 1 6 k 6 n, each k × k principal minor of Mn(a, b, c) has the same value. We find explicit and recursive formulae for the principal minors and the characteristic polynomial of Mn(a, b, c). We also show that all complex polynomials in Mn(a, b, c) are Toeplitz matrices. In particular, the inverse of Mn(a, b, c) is a Toeplitz matrix when it exists.