Noncirculant Toeplitz matrices all of whose powers are Toeplitz

summary:Let $a$, $b$ and $c$ be fixed complex numbers. Let $M_n(a,b,c)$ be the $n\times n$ 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 $1\leq k\leq n$, each $k\times k$ principal minor of $M_n(a,b,c)$ has the same value. We find explicit and recursive formulae for the principal minors and the characteristic polynomial of $M_n(a,b,c)$. We also show that all complex polynomials in $M_n(a,b,c)$ are Toeplitz matrices. In particular, the inverse of $M_n(a,b,c)$ is a Toeplitz matrix when it exists