  A square matrix is said to be circular bidiagonal whenever (i) each nonzero
entry is on the diagonal, or the subdiagonal, or in the top-right corner; (ii)
each subdiagonal entry is nonzero, and the entry in the top-right corner is
nonzero. Let $\mathbb F$ denote a field, and let $V$ denote a nonzero
finite-dimensional vector space over $\mathbb F$. We consider an ordered pair
of $\mathbb F$-linear maps $A: V \to V$ and $A^*: V \to V$ that satisfy the
following two conditions: (i) there exists a basis for $V$ with respect to
which the matrix representing $A$ is circular bidiagonal and the matrix
representing $A^*$ is diagonal; (ii) there exists a basis for $V$ with respect
to which the matrix representing $A^*$ is circular bidiagonal and the matrix
representing $A$ is diagonal. We call such a pair a circular bidiagonal pair on
$V$. We classify the circular bidiagonal pairs up to affine equivalence. There
are two infinite families of solutions, which we describe in detail.
