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65th Workshop on General Algebra, 18th Conference for Young Algebraists
March 21-23, 2003
University of Potsdam
Potsdam, Germany |
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Organizers
Klaus Denecke, Jörg Koppitz
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Permutable rank and its linear preservers
by
Anna Alieva
Moscow State University
Let F be an arbitrary field and Mn(F) a set of n×n-matrices over F.
The theory of linear transformations on Mn(F) preserving different matrix
invariants, relations, or properties has been intensively investigated since 1897,
when Frobenius classified the bijective determinant preservers.
In 1949 Dieudonne proposed a new approach, based on the
fundumental theorem of projective geometry, and received the following
Theorem. All bijective linear transformations T on
matrix algebra Mn(F) preserving the set of all singular matrices are of
standard form: T(X) = PXQ for all matrices X or
T(X) = P(Xt)Q for all matrices X, where Xt denotes the
transposed matrix of X.
The most complete description of the results on Linear Preserver Problems
can be found in the detailed and self-contained surveys [1, 2, 3].
We investigate linear transformations T on
matrix algebra Mn(F) that preserve permutable rank, i.e.
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rk(A1...Ak) = rk(As(1)...As(k)) for any s from Sk |
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implyes
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rk(T(A1)...T(Ak)) = rk(T(As(1))...T(As(k))) for any s from Sk |
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We obtain a classification of the bijective linear preservers of permutable rank:
Theorem. An invertible linear transformation T on matrix algebra
over an arbitrary field F preserves permutable rank if and only if
T(X) = cPXP-1 for all matrices X or T(X) = cP(Xt)P-1 for all
matrices X, where c is in F, matrices P, Q are invertible.
We also give some examples of non-bijective linear preservers of
permutable rank and study linear transformations T on
Mn(F) strongly preserving permutable rank.
[1] Guterman A.E., Mikhalev A.V. General algebra and linear transformations
preserving matrix invariants // Journal of Mathematical Sciences.
To appear.
[2] Li C.-K., Tsing N.K. Linear preserver problems: A brief introduction and
some special techniques // Linear Algebra Appl. 1992. 162-164. 217-235.
[3] Pierce S. and others. A Survey of Linear Preserver Problems // Linear
and Multilinear Algebra. 1992. 33. 1-119.
Date received: December 31, 2002
Copyright © 2002 by the author(s).
The author(s) of this document and the organizers of the conference
have granted their consent to include this abstract in
Atlas Conferences Inc.
Document # cake-11.