Möbius Transformations of Matrix Polynomials

Dedicated to Leiba Rodman on the occasion of his 65th birthday We discuss Möbius transformations for general matrix polynomials over arbitrary fields, analyzing their influence on regularity, rank, determinant, constructs such as com-pound matrices, and on structural features including sparsity and symmetry. Results on the preservation of spectral information contained in elementary divisors, partial multiplic-ity sequences, invariant pairs, and minimal indices are presented. The effect on canonical forms such as Smith forms and local Smith forms, on relationships of strict equivalence and spectral equivalence, and on the property of being a linearization or quadratification are investigated. We show that many important transformations are special instances of Möbius transformations, and analyze a Möbius connection between alternating and palindromic matrix polynomials. Finally, the use of Möbius transformations in solving polynomial inverse eigenproblems is illustrated. Key words. Möbius transformation, generalized Cayley transform, matrix polynomial, matrix pencil