Rank 2 preservers on symmetric matrices with zero trace

Let F be a field, 1 V and 2 V be vector spaces of matrices over F and let  be the rank function. If 1 2 : T V V → is a linear map, and k a fixed positive integer, we say that T is a rank k preserver if for any matrix 1 , A V  ( ) A k  = implies ( ( )) T A k  = . In this paper, we characterize those rank 2 preservers on symmetric matrices with zero trace under certain conditions.

: is a linear map, and k a fixed positive integer, we say that T is a rank k preserver if for any matrix 1 , In this paper, we characterize those rank 2 preservers on symmetric matrices with zero trace under certain conditions.

Introduction
Let nn F  be the algebra of all nn  matrices over a field F . Let () n sl F denote the subspace of nn F  consisting of all matrices with zero trace. In [1], Botta, Pierce and Watkins obtained a useful result concerning the structure of nonsingular linear mapping on () n sl F that preserve nilpotent matrices where F is infinite. In [2], Li and Pierce characterized linear mappings on () n sl F that preserve nonzero nilpotent matrices with rank at most k where k is a fixed positive integer less than n and F is algebraically closed of characteristic zero. Then, Watkins characterized linear mappings from () n sl F to nn F  that preserve rank one matrices where F is an algebraically closed field of characteristic not equal to 2. He applied this result to determine the structure of bilinear mappings on nn F  that have certain rank-preserving properties in [3] and [4] respectively.
Let () n SFbe the vector space of all nn  symmetric matrices over F and 0 ( ( )) n Z S F be its subspace consisting of all symmetric matrices with zero trace. Let 4 n  and F be a field of characteristic greater than 3. Motivated by work of Lim [5] in the characterization of linear rank one preservers on matrices with zero trace, we characterize those rank 2 preservers on symmetric matrices with zero trace under certain conditions in this paper and will discuss some consequences of this characterization in our next paper.

Some definitions and preliminary results
Let U be a vector space over F. We use tensor language in our investigation. This provides us with a larger context. We denote by (2) U the second symmetric product space over U and denoted by ,, x y x y U  , the decomposable elements of (2) U . For each u in U, let 2 u denote uu  . A scalar product on U is a function which assigns a scalar ( , ) x y F  to each ordered pair of vectors , x y U  such that for any ,, x y z U  and any cF  x y y x = We say x is orthogonal to y or x and y are orthogonal if ( ,

ZU
is a subspace of (2) U and we call (2) 0 () ZU the space of traceless 2 nd order symmetric tensors over U.

Proof.
Clearly B is a linearly independent set. Hence it is sufficient to show that B spans . Then Remark. If U is a Euclidean space, then there does not exist any rank 1 vector in 0 ( (2) ).
Let k J denote the set of vectors in (2) U of the form Assume that Then it is clear that there exists From (2) From (3) and (4)

Rank 2 preservers
Let U and W be vector spaces over F . We always assume that U has an orthonormal basis, = and this implies that 12 vv = , since T is a rank 2 preserver. Hence dim

Theorem 3.2 Let
T be a rank 2 preserver from (2) 0 () ZU to (2) W . If dim 4 U  , then one of the following holds: (i) and some one-to-one linear mapping Pf is a second induced power of f such that On the other hand, In view of (5) In view of (6), we have (2) (2) 2 13 23 32 13 32 Likewise, for all distinct ,, i j k . In view of (12) and (7) On the other hand,  The author would like to thank the referees for valuable comments.