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The seminal work by Mackey et al. in 2006 (reference [21] of the article)introduced vector spaces of matrix pencils, with the property that almost allthe pencils in the spaces are strong linearizations of a given square regularmatrix polynomial. This work was subsequently extended by De Ter\'an et al. in2009 (reference [5] of the article) to include the case of square singularmatrix polynomials. We extend this work to non-square matrix polynomials byproposing similar vector spaces of rectangular matrix pencils that are equal tothe ones introduced by Mackey et al. when the polynomial is square. Moreover,the properties of these vector spaces are similar to those in the article by DeTer\'an et al. for the singular case. In particular, the complete eigenvalueproblem associated with the matrix polynomial can be solved by using almostevery matrix pencil from these spaces. Further, almost every pencil in thesespaces can be trimmed to form many smaller pencils that are stronglinearizations of the matrix polynomial which readily solve the completeeigenvalue problem for the polynomial. These linearizations are easier toconstruct and are often smaller than the Fiedler linearizations introduced byDe Ter\'an et al. in 2012 (reference [7] of the article). Further, the globalbackward error analysis by Dopico et al. in 2016 (reference [10] of thearticle) applied to these linearizations, shows that they provide a wide choiceof linearizations with respect to which the complete polynomial eigenvalueproblem can be solved in a globally backward stable manner.

Vector Spaces of Generalized Linearizations for Rectangular Matrix Polynomials