The computation of the Kronecker canonical form of an arbitrary symmetric pencil (Q1110612)

The author considers pencils of \(n\times n\) real symmetric matrices. Two such pencils C-\(\lambda\) B, \(C_ 1-\lambda B_ 1\) are said to be strictly equivalent if there exist non-singular real matrices P, Q such that \(P(C-\lambda B)Q=C_ 1-\lambda B_ 1.\) Kronecker gave canonical forms for symmetric pencils under strict equivalence, and it is the author's aim to devise practical methods for computing them. A sharp distinction is to be drawn between the regular case, where det(C- \(\lambda\) B)\(\not\equiv 0\), and the contrary singular case. A practical difficulty in the latter is that slight perturbations may render the pencil regular, thus radically altering the canonical form (which is a direct sum of lower-dimensional pencils, some regular and others singular). The author first gives a detailed description of an algorithm for calculating the canonical form of a regular pencil. It has the advantage over the method of \textit{P. Van Dooren} [ibid. 27, 103-140 (1979; Zbl 0416.65026)] of preserving symmetry and is claimed to be quicker. A more general algorithm is then described for computing that regular part of the canonical form of a singular pencil, which corresponds to the finite elementary divisors. The author points to several advantages of his method, while acknowledging that the fundamental difficulty mentioned above remains.