Symbolic Computation of the Aluthge Transform

The algorithm for the symbolic computation of the Aluthge transform of a polynomial matrix is derived in this paper. For this purpose, the well-known PSVD by PQRD algorithm is considered to avoid square roots of polynomials in the Aluthge transform matrix. The algorithm for the symbolic computation of the polar decomposition for polynomial matrices is developed as well. Furthermore, the Aluthge transform for rank-one matrices defined by cross-products of vectors is symbolically calculated and presented in a closed formula. © 2017, Springer International Publishing

Representations of generalized inverses via full-rank QDR decomposition

In this paper, novel representations of generalized inverses of rational matrices are developed. Therefore, a unified approach for the computation of 1,2,3 and 1,2,4 inverses and Moore-Penrose inverse of a given matrix A is considered. Full-rank QDR decomposition of a rational matrix is utilized to avoid the square roots of rational expressions in the evaluations, making the given algorithm very suitable for symbolic computations of generalized matrix inverses. Furthermore, we developed an algorithm for symbolic computation of the Moore-Penrose inverse of a polynomial matrix using the full-rank QDR decomposition, therefore maximizing the potential of using square root–free polynomial entries. Introduced algorithms are illustrated via numerical examples. © 2020, Springer Science+Business Media, LLC, part of Springer Nature