The algebraic structure of pencils and block Toeplitz Matrices

We restate several results on the Kronecker structure obtainable after a given matrix pencil is affected by an arbitrarily small perturbation, and state some new results on the Kronecker structure after a new row is appended or a column deleted. The results are simple consequences of a theory of majorization for semi-infinite integer sequences. The bounds after a new row is appended correspond to bounding the controllability indices after adding a single input to a linear time-invariant dynamical system. We illustrate the known relations between the Kronecker indices, the Segre characteristics, and the nullities of certain block Toeplitz matrices constructed from the original pencils and discuss how those nullities change when the pencil is affected by a perturbation or a new row.