Polynomial Identity Rings

Abstract. In this paper we provide generalizations of Specht's Theorem which states that two n × n matrices A and B are unitarily equivalent if and only if all traces of words in two non-commuting variables applied to the pairs (A, A * ) and (B, B * ) coincide. First, we obtain conditions which allow us to extend this to simultaneous similarity or unitary equivalence of families of operators, and secondly, we show that it suffices to consider a more restricted family of functions when comparing traces. Our results do not require the traces of words in (A, A * ) and (B, B * ) to coincide, but only to be close.

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“…The proof, however, is somewhat involved, and will not be elaborated here. We direct the interested reader to Chapter 6 of [1])…”

Section: Introductionmentioning

confidence: 99%

An approximate, multivariable version of Specht's theorem

Marcoux

Mastnak

Radjavi

2007

Linear and Multilinear Algebra

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“…Consider the natural GL n -action on the space V = M r n of r-tuples of n×n-matrices by simultaneous conjugation. The variety X = M r n // GL n has been extensively studied in the context of both invariant and PI theories; an overview of this research area can be found in [6], [21] or [5]. In [22,23] the second author constructed a large family of automorphisms of X = V // PGL n (for r n + 1).…”

Section: Then the Luna Stratification In V // G Is Intrinsic Moreovementioning

confidence: 99%

Is the Luna stratification intrinsic?

Kuttler¹,

Reichstein²

2008

Ann. inst. Fourier

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Let G → GL(V ) will be a finite-dimensional linear representation of a reductive linear algebraic group G on a finite-dimensional vector space V , defined over an algebraically closed field of characteristic zero. The categorical quotient V // G carries a natural stratification, due to D. Luna. This paper addresses the following questions:(i) Is the Luna stratification of X intrinsic? That is, does every automorphism of V // G map each stratum to another stratum?(ii) Are the individual Luna strata in X intrinsic? That is, does every automorphism of V // G maps each stratum to itself?In general, the Luna stratification is not intrinsic. Nevertheless, we give positive answers to questions (i) and (ii) for large classes of interesting representations.

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“…Important properties of matrix operations are often expressed in form of laws or identities such as the associative law for multiplication of matrices. Studying matrix identities that involve multiplication and addition is a classic research direction that was motivated by several important problems in geometry and algebra (see [1] for a survey of the origins of the theory) and that has eventually led to the profound and beautiful theory of PI-rings [14,21,23,47]. Matrix identities involving along with multiplication and addition also certain involution operations (such as taking the usual or symplectic transpose of a matrix) have attracted much attention as well, see, for instance, [9,10,15,20,47].…”

Section: Matricesmentioning

confidence: 99%