Similarity of non-everywhere defined linear maps
The equivalence between linear maps defined on a subspace is studied and applied to block similarity of rectangular matrices. In that way, the authors find the complete families of invariants obtained by \textit{P. Brunovsky} [Kibernetika, Praha 6, 173-188 (1970; Zbl 0199.482)] and \textit{I. Zaballa} [Linear Algebra Appl. 87, 113-146 (1987; Zbl 0632.15003)], and the results of \textit{I. Gohberg}, \textit{M. A. Kaashoek} and \textit{F. van Schagen} [Integral Equations Oper. Theory 3, 350-396 (1980; Zbl 0443.15005)] and of \textit{I. Gohberg}, \textit{P. Lancaster}, and \textit{L. Rodman} [Invariant subspaces of matrices with applications (1986; Zbl 0608.15004), Chapter 6]) for this particular case. Also, the authors describe explicit methods for obtaining the reduced matrix and the corresponding bases. Section I. ``Classification of linear maps defined on a subspace'' contains linear maps defined on a subspace, equivalence relations, direct invariant subspaces, the key theorem, the induced maps and the classification theorem. Section II is reduced canonical matrix of a linear map defined on a subspace. Section III. ``Computation on the invariants'' contains determination of the subspaces, computation of \(k\), characterization of the particular case, computation of the similarity invariants and an example. Section IV and V are ``The dual case: linear maps defined modulo a subspace,'' and ``Block similarity of matrices''.
block similarity of rectangular matrices
scientific article; zbMATH DE number 57561