Generalized invariant polynomials and the generalized companion form

TL;DR: In this paper, a new canonical form for regular pencils of matrices was constructed with generalized companion blocks, and it was shown that every regular pencil is strictly equivalent to such a canonical form.

Abstract: In the last two decades Soviet m athem aticians have produced a series of rem arkable books, whose common feature is the stress laid on thoroughness and intelligibility ra the r than on slickness of presentation. Professor G antm acher’s two volumes fall into this category, and readers in English-speaking countries owe a debt of g ratitude to the translator and the publishers for making available to them this im portant treatise. The work provides not so much a system atic development of linear algebra as a detailed study of a num ber of special topics in m atrix theory, an account of some of which is not easily found elsewhere. However, the discussion is based firmly on the general theory of matrices and quadratic forms. The trea tm en t given here is largely in the spirit of m odern linear algebra, and particular atten tion is paid to the ‘geom etric’ theory of elem entary divisors and to decompositions of a vector space into sub­ spaces invariant w ith respect to a given linear operator. The remaining parts of the book, comprising about eight of the fifteen chapters, are reserved for the investigation of more special topics, of which we m ention the following: (i) functions of a m atrix (defined in term s of the LagrangeSylvester interpolation); (ii) m atrix equations; (iii) singular pencils; (iv) non-negative matrices, stochastic m atrices, Markov chains; (v) systems of linear differential equations; (vi) the algorithm of RouthH urw itz; (vii) H ankel matrices. Even this brief summ ary gives some indication of the formidable array of problems deployed w ith painstaking thoroughness by Professor Gantm acher. Indeed, he does not shrink from lengthy and involved argum ents of a kind more commonly found in research papers than in books. However, the m aterial discussed is so interesting and the presentation so clear th a t the book is sure of a welcome among a wide circle of readers. L. Mir s k y

TL;DR: In this paper, a new canonical form for regular pencils of matrices was constructed with generalized companion blocks, and it was shown that every regular pencil is strictly equivalent to such a canonical form.

Abstract: In the last two decades Soviet m athem aticians have produced a series of rem arkable books, whose common feature is the stress laid on thoroughness and intelligibility ra the r than on slickness of presentation. Professor G antm acher’s two volumes fall into this category, and readers in English-speaking countries owe a debt of g ratitude to the translator and the publishers for making available to them this im portant treatise. The work provides not so much a system atic development of linear algebra as a detailed study of a num ber of special topics in m atrix theory, an account of some of which is not easily found elsewhere. However, the discussion is based firmly on the general theory of matrices and quadratic forms. The trea tm en t given here is largely in the spirit of m odern linear algebra, and particular atten tion is paid to the ‘geom etric’ theory of elem entary divisors and to decompositions of a vector space into sub­ spaces invariant w ith respect to a given linear operator. The remaining parts of the book, comprising about eight of the fifteen chapters, are reserved for the investigation of more special topics, of which we m ention the following: (i) functions of a m atrix (defined in term s of the LagrangeSylvester interpolation); (ii) m atrix equations; (iii) singular pencils; (iv) non-negative matrices, stochastic m atrices, Markov chains; (v) systems of linear differential equations; (vi) the algorithm of RouthH urw itz; (vii) H ankel matrices. Even this brief summ ary gives some indication of the formidable array of problems deployed w ith painstaking thoroughness by Professor Gantm acher. Indeed, he does not shrink from lengthy and involved argum ents of a kind more commonly found in research papers than in books. However, the m aterial discussed is so interesting and the presentation so clear th a t the book is sure of a welcome among a wide circle of readers. L. Mir s k y

TL;DR: In this article, the Routh-Hurwitz problem of singular pencils of matrices has been studied in the context of systems of linear differential equations with variable coefficients, and its applications to the analysis of complex matrices have been discussed.