Sylvester's Identity Research Articles

A Triangle Algorithm of Padé-Type Approximant for Two-Dimensional Fredholm Integral Equations of the Second Kind

The function-valued Padé-type approximation (2DFPTA) is used to solve two-dimensional Fredholm integral equation of the second kind. In order to compute 2DFPTA, a triangle recursive algorithm based on Sylvester identity is proposed. The advantage of this algorithm is that, in the process of calculating 2DFPTA to avoid the calculation of the determinant, it can start from the initial value, from low to high order, and gradually proceeds. Compared with the original two methods, the numerical examples show that the algorithm is effective.

Representations of Generalized Inverses and Drazin Inverse of Partitioned Matrix with Banachiewicz-Schur Forms

Representations of 1,2,3-inverses, 1,2,4-inverses, and Drazin inverse of a partitioned matrix M=ABCD related to the generalized Schur complement are studied. First, we give the necessary and sufficient conditions under which 1,2,3-inverses, 1,2,4-inverses, and group inverse of a 2×2 block matrix can be represented in the Banachiewicz-Schur forms. Some results from the paper of Cvetković-Ilić, 2009, are generalized. Also, we expressed the quotient property and the first Sylvester identity in terms of the generalized Schur complement.

Can Sylvester's determinantal identity, equivalently Muir's law of extensible minors be generalized?

We show that Sylvester's classical determinantal identity is equivalent to its generalizations given by the Mühlbach–Gasca–(López-Carmona)–Ramírez identity and the Beckermann–Mühlbach identity. We also show that the generating extension principles associated with each of these three identities (Muir's law of extensible minors, Mühlbach's extension principle and the extension principle given in Camargo, 2013 [3]) are equivalent and we use this fact to conclude that all these six statements (and also Jacobi's identity on the minors of the adjugate and Cayley's law of complementaries) are actually equivalent. These findings lead naturally to enquire whether there would exist a generalization of Sylvester's identity (or an extension principle) which could not be derived from Sylvester's identity itself.

Two expansion formulas involving the Rogers–Szegő polynomials with applications

Using two expansion formulas for the Rogers–Szegő polynomials and the Stieltjes–Wigert polynomials, we give new proofs of a variety of important classical formulas including Bailey's6ψ6summation, the Askey–Wilson integral and its extension. Furthermore, we give nontrivial extensions of the Andrews multiple version of the Rogers–Selberg identity, as well as the Sylvester identity.

Contiguous relations and summation and transformation formulae for basic hypergeometric series

By using contiguous relations for basic hypergeometric series, we give simple proofs of Bailey's summation, Carlitz's summation, Sears' to transformation, Sears' transformations, Chen's bibasic summation, Gasper's split poised transformation and Chu's bibasic symmetric transformation. Along the same line, finite forms of Sylvester's identity, Jacobi's triple product identity and Kang's identity are also obtained.

A NEW EXPLORATION OF THE LEBESGUE IDENTITY

We introduce a new combinatorial proof of the Lebesgue identity which allows us to find a new finite form of the identity. Using this new finite form we are able to make new observations about special cases of the Lebesgue identity, namely the "little" Göllnitz theorems and Sylvester's identity.

The Hankel pencil conjecture

The Toeplitz pencil conjecture stated in [W. Schmale, P.K. Sharma, Problem 30-3: singularity of a toeplitz matrix, IMAGE 30 (2003); W. Schmale, P.K. Sharma, Cyclizable matrix pairs over C [ x ] and a conjecture on toeplitz pencils, Linear Algebra Appl. 389 (2004) 33-42] is equivalent to a conjecture for n × n Hankel pencils of the form H n ( x ) = ( c i + j - n + 1 ) , where c 0 = x is an indeterminate, c l = 0 for l < 0 , and c l ∈ C ∗ = C ⧹ { 0 } , for l ⩾ 1 . In this paper it is shown to be implied by another conjecture, which we call the root conjecture. The root conjecture asserts a strong relationship between the roots of certain submaximal minors of H n ( x ) specialized to have c 1 = c 2 = 1 . We give explicit formulae in the c i for these minors and prove the root conjecture for minors m nn , m n - 1 , n of degree ⩽ 6 . This implies the Hankel Pencil conjecture for matrices up to size 8 × 8 . The main tools involved are a partial parametrization of the set of solutions of systems of polynomial equations that are both homogeneous and index sum homogeneous, and use of the Sylvester identity for matrices.

Non-Commutative Sylvester's Determinantal Identity

Sylvester's identity is a classical determinantal identity with a simple linear algebra proof. We present combinatorial proofs of several non-commutative extensions, and find a $\beta$-extension that is both a generalization of Sylvester's identity and the $\beta$-extension of the quantum MacMahon master theorem.

A Generalized Sylvester Identity and Fraction-free Random Gaussian Elimination

Sylvester’s identity is a well-known identity that 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.

On vector Hankel determinants

Recently, a definition of Hankel determinants H k n whose entries belong to a real finite dimensional linear space R d has been given. This definition is based on designants and Clifford algebra. Such determinants appear in the theory of vector orthogonal polynomials, vector Padé approximants, in the algebraic approach to the vector ε-algorithm and other areas. Its fundamental algebraic property is that it is a vector of the real linear space R d . Sylvester's identity is still valid for computing recursively these determinants, involving elements of Clifford algebra. The aim of this paper is to show that this way (Sylverster's identity) is not an optimal one and to propose a more effecient alternative one, since it avoids the use of the Clifford algebra structure. This new identity will be also called Sylvester's identity since it is equivalent to the classical Sylvester's identity in the scalar case. It allows us also to recover the fundamental property more easily. Moreover, an expression of H k n in terms of classical determinants will be given and also some new determinantal identities.

A generalization of Sylvester's identity

We consider a new generalization of Euler's and Sylvester's identities for partitions. Our proof is based on an explicit bijection.

A recursive family of differential polynomials generated by the sylvester identity and addition theorems for hyperelliptic Kleinian functions

A recursive family of differential polynomials generated by the sylvester identity and addition theorems for hyperelliptic Kleinian functions

Matrix extrapolation algorithms

Extrapolation algorithms are used for accelerating scalar or vector sequences. They are also used for solving systems of linear and nonlinear equations. These algorithms are expressed in terms of a ratio of two determinants, as the E-algorithm and the general recursive projection algorithm ( grpa). In this paper we define the matrix extrapolation problem and we use the Schur complement and the Sylvester identity for solving this problem; we give two transformations, equivalent to the E-algorithm and the grpa, for the matrix case. We give also a characterization of their kernels.

Some properties of the recursive projection and interpolation algorithms

The recursive projection algorithm (rpa), derived from the vector Sylvester identity, has a connection with the recursive interpolation algorithm (ria). These algorithms have many applications and connections with some other methods used in various areas of numerical analysis. The aim of this paper is to study some properties of these algorithms. We use properties of projectors, a matrix implementation and the Schur complement extended to the vector case, to give a result about their finite termination.

On extending determinantal identities

A new principle for extending determinantal identities is established which generalizes Muir's classical law of extensible minors. The proof makes use of general elimination strategies and of generalized Schur complements. This principle allows either the list of columns or the list of rows extending the corresponding list in the given identity to depend on the latter. As applications of this technique Karlin's and a generalization of Sylvester's identity are derived.

A generalization of Sylvester's identity on determinants and some applications

A generalization of Sylvester's identity on determinants is proved by elimination techniques. As an application some inequalities for determinants of totally positive or positive definite matrices are derived.

Other determinental conditions for concavity and quasi-concavity

In the case of twice differentiable functions, the most usual tests of concavity and quasi- concavity are those concerning the monotonicity (with respect to l) property of the signs of the lth principal minors or the lth principal bordered minors of the hessian matrix. These tests are irreducible one with the other. However, we show that the symmetry of the procedures and the relationship between principal and principal bordered minors provides the possibility of a bridge between them and then an appreciable reduction of computation. Such a bridge is offered by the Sylvester identity. Then, new determinental criteria of concavity and quasi-concavity follow.

The mühlbach-neville-aitken algorithm and some extensions

A new proof of the Mühlbach-Neville-Aitken algorithm for interpolation by a linear family of functions forming a Chebyshev system is given. This proof is based on Sylvester's identity for determinants. The algorithm is then applied to the general interpolation problem, and applications to orthogonal polynomials and Padé-type approximants are treated. Finally the extension to rational interpolation is also studied.