Elementary matrix

Summary

In mathematics, an elementary matrix is a matrix which differs from the identity matrix by one single elementary row operation. The elementary matrices generate the general linear group GLn(F) when F is a field. Left multiplication (pre-multiplication) by an elementary matrix represents elementary row operations, while right multiplication (post-multiplication) represents elementary column operations. Elementary row operations are used in Gaussian elimination to reduce a matrix to row echelon form. They are also used in Gauss–Jordan elimination to further reduce the matrix to reduced row echelon form. Elementary row operations There are three types of elementary matrices, which correspond to three types of row operations (respectively, column operations): ;Row switching: A row within the matrix can be switched with another row. : R_i \leftrightarrow R_j ;Row multiplication: Each element in a row can be multiplied by a non-zero constant. It is also known as

Official source

https://en.wikipedia.org/wiki/Elementary_matrix

About this result

This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.

Related publications (4)

Related publications

Related people

Related units

Related concepts (15)

Related concepts

Related courses (13)

Related courses

Related lectures (49)

Related lectures

This work describes a fast fully homomorphic encryption scheme over the torus (TFHE) that revisits, generalizes and improves the fully homomorphic encryption (FHE) based on GSW and its ring variants. The simplest FHE schemes consist in bootstrapped binary gates. In this gate bootstrapping mode, we show that the scheme FHEW of Ducas and Micciancio (Eurocrypt, 2015) can be expressed only in terms of external product between a GSW and an LWE ciphertext. As a consequence of this result and of other optimizations, we decrease the running time of their bootstrapping from 690 to 13 ms single core, using 16 MB bootstrapping key instead of 1 GB, and preserving the security parameter. In leveled homomorphic mode, we propose two methods to manipulate packed data, in order to decrease the ciphertext expansion and to optimize the evaluation of lookup tables and arbitrary functions in RingGSW-based homomorphic schemes. We also extend the automata logic, introduced in Gama et al. (Eurocrypt, 2016), to the efficient leveled evaluation of weighted automata, and present a new homomorphic counter called TBSR, that supports all the elementary operations that occur in a multiplication. These improvements speed up the evaluation of most arithmetic functions in a packed leveled mode, with a noise overhead that remains additive. We finally present a new circuit bootstrapping that converts LWE ciphertexts into low-noise RingGSW ciphertexts in just 137 ms, which makes the leveled mode of TFHE composable and which is fast enough to speed up arithmetic functions, compared to the gate bootstrapping approach. Finally, we provide an alternative practical analysis of LWE based schemes, which directly relates the security parameter to the error rate of LWE and the entropy of the LWE secret key, and we propose concrete parameter sets and timing comparison for all our constructions.

SPRINGER2020

Let A be a d-dimensional smooth algebra over a perfect field of characteristic not 2. Let Um(n+1)(A)/En+1 (A) be the set of unimodular rows of length n + 1 up to elementary transformations. If n >= (d + 2)/2, it carries a natural structure of group as discovered by van der Kallen. If n = d >= 3, we show that this group is isomorphic to a cohomology group H-d (A, G(d+1)). This extends a theorem of Morel, who showed that the set Um(d+1)(A)/SLd+1(A) is in bijection with H-d (A, G(d+1))/SLd+1(A). We also extend this theorem to the case d = 2. Using this, we compute the groups Um(d+1)(A)/Ed+1(A) when A is a real algebra with trivial canonical bundle and such that Spec(A) is rational. We then compute the groups Um(d+1)(A)/SLd+1 (A) when d is even, thus obtaining a complete description of stably free modules of rank d on these algebras. We also deduce from our computations that there are no stably free non free modules of top rank over the algebraic real spheres of dimension 3 and 7.

2011

On finitely generated modules over quasi-Euclidean rings

Luc Guyot

Let R be a unital commutative ring, and let M be an R-module that is generated by k elements but not less. Let be the subgroup of generated by the elementary matrices. In this paper we study the action of by matrix multiplication on the set of unimodular rows of M of length . Assuming R is moreover Noetherian and quasi-Euclidean, e.g., R is a direct product of finitely many Euclidean rings, we show that this action is transitive if . We also prove that is equipotent with the unit group of where is the first invariant factor of M. These results encompass the well-known classification of Nielsen non-equivalent generating tuples in finitely generated Abelian groups.

Springer Verlag2017