Permutable rank and its linear preservers
by
Anna Alieva
Moscow State University

Let F be an arbitrary field and M_n(F) a set of n×n-matrices over F. The theory of linear transformations on M_n(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 M_n(F) preserving the set of all singular matrices are of standard form: T(X) = PXQ for all matrices X or T(X) = P(X^t)Q for all matrices X, where X^t 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 M_n(F) that preserve permutable rank, i.e.

rk(A1...Ak) = rk(As(1)...As(k)) for any s from Sk

implyes

rk(T(A1)...T(Ak)) = rk(T(As(1))...T(As(k))) for any s from Sk

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(X^t)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 M_n(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.