On the minimal ranks of matrix pencils and the existence of a best approximate block-term tensor decomposition

Under the action of the general linear group, the ranks of matrices A and B forming a m × n pencil A+ λB can change, but in a restricted manner. Specifically, to every pencil one can associate a pair of minimal ranks, which is unique up to a permutation. This notion can be defined for matrix pencils and, more generally, also for matrix polynomials of arbitrary degree. The natural hierarchy it induces in a pencil space is discussed. Then, a characterization of the minimal ranks of a pencil in terms of its Kronecker canonical form is provided. We classify the orbits according to their minimal ranks—under the action of the general linear group—in the case of real pencils with m,n ≤ 4. By relying on this classification, we show that no real regular 4 × 4 pencil having only complex-valued eigenvalues admits a best approximation (in the norm topology) on the set of real pencils whose minimal ranks are bounded by 3. These non-approximable pencils form an open set, which is therefore of positive volume. Our results can be interpreted from a tensor viewpoint, where the minimal ranks of a degree-(d− 1) matrix polynomial characterize the minimal ranks of matrices constituting a block-term decomposition of a m × n × d tensor into a sum of matrix-vector tensor products.