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Linear preservers for matrix inequalities and partial orderings. (English) Zbl 0985.15018

Let \(M_{n}(F)\) denote the algebra of \(n\times n\) matrices over the field \(F\). The author calls a linear transformation \(T\) of \(M_{n}(F)\) into itself of “standard form” if \(T\) either has the form \(X\mapsto AXB\) or the form \(X\mapsto AX^{T}B\) for some fixed invertible matrices \(A\) and \(B\). There have been many papers by a series of authors in which it has been shown that transformations \(T\) which preserve a wide variety of properties of \(M_{n}(F)\) are necessarily of standard form [see, for example, C.-K. Li and N.-K. Tsing, Linear Algebra Appl. 162-164, 217-235 (1992; Zbl 0762.15016)].
In the present paper the author considers the partial ordering \(\preceq\) on \(M_{n}(F)\) defined by \(A\preceq B\) when \(rank(B-A)=rank(B)-rank(A).\) (This is called the rank-subtractivity ordering and is equivalent to the minus ordering). He shows that, provided \(\left|F\right|>n\), every invertible linear transformation \(T\) of \(M_{n}(F)\) which preserves \(\preceq\) is of standard form, and also proves a number of related results.
Reviewer: J.D.Dixon (Ottawa)

MSC:

15B48 Positive matrices and their generalizations; cones of matrices
15A04 Linear transformations, semilinear transformations
15A45 Miscellaneous inequalities involving matrices
15A30 Algebraic systems of matrices

Citations:

Zbl 0762.15016
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References:

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