Various proofs of Sylvester's (determinant) identity

doi:10.1016/S0378-4754(96)00035-3

Mathematics and Computers in Simulation

Volume 42, Issues 4–6, November 1996, Pages 585-593

https://doi.org/10.1016/S0378-4754(96)00035-3 Get rights and content

Abstract

Despite the fact that the importance of Sylvester's determinant identity has been recognized in the past, we were able to find only one proof of it in English (Bareiss, 1968), with reference to some others. (Recall that Sylvester (1857) stated this theorem without proof.) Having used this identity, recently, in the validity proof of our new, improved, matrix-triangularization subresultant polynomial remainder sequence method (Akritas et al., 1995), we decided to collect all the proofs we found of this identity-one in English, four in German and two in Russian, in that order-in a single paper (Akritas et al., 1992). It turns out that the proof in English is identical to an earlier one in German. Due to space limitations two proofs are omitted.

References (10)

A.G. Akritas et al.
Computation of polynomial remainder sequences in integral domains
Reliable Computing
(1995)
A.G. Akritas et al.
Various proofs of Sylvester's (determinant) identity
E.H. Bareiss
Sylvester's identity and multistep integer preserving Gaussian elimination
Math. Comp.
(1968)
R.A. Horn et al.
Topics in Matrix Analysis
(1991)
C.G.J. Jacobi
Ueber die Bildung und die Eigenschaften der Determinanten

There are more references available in the full text version of this article.

Cited by (67)

Vision-based navigation in low Earth orbit – Using the stars and horizon as an alternative PNT
2023, Advances in Space Research
The risk to low Earth orbit space assets has never been greater than in the 2020s. Recent conflicts have led many governments, militaries, and industries to call for greater mission assurance, increased resilience and redundancy, and to make space sustainable for the future. As part of these urgencies is a growing need for autonomous PNT that does not rely on exterior infrastructure. The work disclosed details models and illustrates the potential for using visual-based navigation to deliver this alternative assurance for space domain users. Approaches from lunar navigation are adopted for orbits with closer proximity to Earth, where using known orbit mechanic models the user position can be calculated and predicted with strong performance (100 m). This is demonstrated both in a lost-in-space scenario and in situations of denied service. This system capability may also be adopted for future telecommunication mega-constellations to deliver a ranging signal for Earth users, known as LEO PNT.
Critical groups of arithmetical structures under a generalized star-clique operation
2023, Linear Algebra and Its Applications
An arithmetical structure on a finite, connected graph without loops is given by an assignment of positive integers to the vertices such that, at each vertex, the integer there is a divisor of the sum of the integers at adjacent vertices, counted with multiplicity if the graph is not simple. Associated to each arithmetical structure is a finite abelian group known as its critical group. Keyes and Reiter gave an operation that takes in an arithmetical structure on a finite, connected graph without loops and produces an arithmetical structure on a graph with one fewer vertex. We study how this operation transforms critical groups. We bound the order and the invariant factors of the resulting critical group in terms of the original arithmetical structure and critical group. When the original graph is simple, we determine the resulting critical group exactly.
A reduction algorithm for reconstructing periodic Jacobi matrices in Minkowski spaces
2022, Applied Mathematics and Computation
The periodic Jacobi inverse eigenvalue problem concerns the reconstruction of a periodic Jacobi matrix from prescribed spectral data. In Minkowski spaces, with a given signature operator $H = diag (1, 1, \dots, 1, - 1)$ , the corresponding matrix is a periodic pseudo-Jacobi matrix. The inverse eigenvalue problem for such matrices consists in the reconstruction of pseudo-Jacobi matrices, with the same order and signature operator $H$ . In this paper we solve this problem by applying Sylvester’s identity and Householder transformation. The solution number and the corresponding reconstruction algorithm are here exhibited, and illustrative numerical examples are given. Comparing this approach with the known Lanczos algorithm for reconstructing pseudo-Jacobi matrices, our method is shown to be more stable and effective.
An Introduction to T-Systems - with a special Emphasis on Sparse Moment Problems, Sparse Positivstellensätze, and Sparse Nichtnegativstellensätze
2024, arXiv
Distributed Experimental Design Networks
2024, arXiv
Singularities of determinantal pure pairs
2023, Bollettino dell'Unione Matematica Italiana

View all citing articles on Scopus

¹: Partially supported by GRF grant 3089-XX-0038 (1993) of the University of Kansas.

View full text

Various proofs of Sylvester's (determinant) identity

Abstract

Computation of polynomial remainder sequences in integral domains

Reliable Computing

Various proofs of Sylvester's (determinant) identity

Sylvester's identity and multistep integer preserving Gaussian elimination

Math. Comp.

Topics in Matrix Analysis

Ueber die Bildung und die Eigenschaften der Determinanten