On the minimal ranks of matrix pencils and the existence of a best approximate block-term tensor decomposition
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Under the action of the general linear group, the ranks of matrices A and B forming a m × n pencil A + λ B 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.
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