  Schur decompositions and the corresponding Schur forms of a single matrix, a
pair of matrices, or a collection of matrices associated with the periodic
eigenvalue problem are frequently used and studied. These forms are
upper-triangular complex matrices or quasi-upper-triangular real matrices that
are equivalent to the original matrices via unitary or, respectively,
orthogonal transformations. In general, for theoretical and numerical purposes
we often need to reduce, by admissible transformations, a collection of
matrices to the Schur form. Unfortunately, such a reduction is not always
possible. In this paper we describe all collections of complex (real) matrices
that can be reduced to the Schur form by the corresponding unitary (orthogonal)
transformations and explain how such a reduction can be done. We prove that
this class consists of the collections of matrices associated with pseudoforest
graphs. In other words, we describe when the Schur form of a collection of
matrices exists and how to find it.
