Non-commutative Sylvester's determinantal identity

Sylvester's identity is a classical determinantal identity with a straightforward linear algebra proof. We present a new, combinatorial proof of the identity, prove several non-commutative versions, and find a $\beta$-extension that is both a generalization of Sylvester's identity and the $\beta$-extension of the MacMahon master theorem.Comment: 28 pages, 8 figure

A generalized Sylvester identity and fraction-free random Gaussian elimination

Sylvester's identity is a well-known identity which can be used to prove that certain Gaussian elimination algorithms are fraction-free. In this paper we will generalize Sylvester's identity and use it to prove that certain random Gaussian elimination algorithms are fraction-free. This can be used to yield fraction-free algorithms for solving Ax = b (x 0) and for the simplex method in linear programming

Subresultants, sylvester sums and the rational interpolation problem

We present a solution for the classical univariate rational interpolation problem by means of (univariate) subresultants. In the case of Cauchy interpolation (interpolation without multiplicities), we give explicit formulas for the solution in terms of symmetric functions of the input data, generalizing the well-known formulas for Lagrange interpolation. In the case of the osculatory rational interpolation (interpolation with multiplicities), we give determinantal expressions in terms of the input data, making explicit some matrix formulations that can independently be derived from previous results by Beckermann and Labahn.Fil: D'Andrea, Carlos. Universidad de Barcelona; EspañaFil: Krick, Teresa Elena Genoveva. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santaló". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santaló"; ArgentinaFil: Szanto, Agnes. North Carolina State University; Estados Unido

An algorithm for the quadratic approximation

The quadratic approximation is a three dimensional analogue of the two dimensional Pade approximation. A determinantal expression for the polynomial coefficients of the quadratic approximation is given. A recursive algorithm for the construction of these coefficients is derived. The algorithm constructs a table of quadratic approximations analogous to the Pade table of rational approximations

An algorithm for the quadratic approximation

The quadratic approximation is a three dimensional analogue of the two dimensional Pade approximation. A determinantal expression for the polynomial coefficients of the quadratic approximation is given. A recursive algorithm for the construction of these coefficients is derived. The algorithm constructs a table of quadratic approximations analogous to the Pade table of rational approximations

A restricted signature normal form for Hermitian matrices, quasi-spectral decompositions, and applications

In recent years, a number of results on the relationships between the inertias of Hermitian matrices and the inertias of their principal submatrices appeared in the literature. We study restricted congruence transformation of Hermitian matrices M which, at the same time, induce a congruence transformation of a given principal submatrix A of M. Such transformations lead to concept of the restricted signature normal form of M. In particular, by means of this normal form, we obtain short proofs of most of the known inertia theorems and also derive some new results of this type. For some applications, a special class of almost unitary restricted congruence transformations turns out to be useful. We show that, with such transformations, M can be reduced to a quasi-diagonal form which, in particular, displays the eigenvalues of A. Finally, applications of this quasi-spectral decomposition to generalize inverses and Hermitian matrix pencils are discussed