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

Zbl 0762.15016
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References:

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