17B01 — 2 articles found.

Representation of a Zero Trace Matrix as a Commutator

C. R. Math. Rep. Acad. Sci. Canada Vol. 46 (2) 2024, pp. 41–45
Vol.46 (2) 2024
Ry Cyna; Irwin S. Pressman Details
(Received: 2024-05-01 , Revised: 2024-05-22 )
(Received: 2024-05-01 , Revised: 2024-05-22 )

Ry Cyna, Department of Physics, University of Toronto, McLennan Physical Laboratories, 60 St George St., Toronto, Ontario, CANADA M5S 1A7; e-mail: ry.cyna@mail.utoronto.ca

Irwin S. Pressman, Fields Institute for Research in Mathematical Sciences, 222 College St., Toronto, Ontario,
CANADA M5S 1A7 and School of Mathematics and Statistics, Carleton University, 1125 Colonel By Drive, Ottawa, Ontario, CANADA K1S 5B6; e-mail: irwinpressman@cunet.carleton.ca

Abstract/Résumé:

A new solution to the problem of representing a zero-trace matrix as a commutator of a pair of matrices is presented. For diagonalizable matrices, the solution first consists of a Toeplitz matrix \(H\) with \(1\) on the superdiagonal, and a second matrix with cumulative sums of eigenvalues on the subdiagonal. Defective matrices use the Jordan Normal Form to add cumulative sums of the ones and zeros in the Jordan superdiagonal to the diagonal of the second matrix. We show that every matrix is a polynomial in \(H\) and its transpose.

Une nouvelle solution au problème de la représentation d’une matrice sans trace comme commutateur d’une paire de matrices est présentée. Pour les matrices diagonalisables, la solution consiste d’abord en une matrice de Toeplitz \(H\) avec \(1\) sur le superdiagonale, et une deuxième matrice avec des sommes cumulées de valeurs propres sur la sous-diagonale. Les matrices défectueuses utilisent la forme normale de Jordan pour ajouter les sommes cumulèes des uns et des zéros de la superdiagonale de Jordan à la diagonale de la deuxième matrice. Nous montrons que toute matrice est un polynôme dans \(H\) et sa transposée.

Keywords: Toeplitz matrices, Zero trace matrix, commutator, matrix exponential

AMS Subject Classification: Matrix equations and identities, Identities; free Lie (super)algebras 15A24, 17B01

PDF(click to download): Representation of a Zero Trace Matrix as a Commutator

Line bundles and conjugacy theorems for toroidal Lie algebras

C. R. Math. Rep. Acad. Sci. Canada Vol. 22 (3) 2000, pp. 125–128
Vol.22 (3) 2000
A. Pianzola Details
(Received: 1999-11-11 )
(Received: 1999-11-11 )

A. Pianzola

Abstract/Résumé:

No abstract available but the full text pdf may be downloaded at the title link below.

Keywords:

AMS Subject Classification: Group schemes, Identities; free Lie (super)algebras, Infinite-dimensional Lie groups and their Lie algebras 14L15, 17B01, 22E65

PDF(click to download): Line bundles and conjugacy theorems for toroidal Lie algebras

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