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Linear operators that preserve pairs of matrices which satisfy extreme rank properties. (English) Zbl 1004.15025

Let \({\mathbb F}\) be a field with at least \(m+2\) elements and of characteristic not \(2\). Then a linear operator \(T:M_{m,n}({\mathbb F})\to M_{m,n}({\mathbb F})\) preserves the set of all rank-sum-maximal (rank-sum-minimal) pairs of matrices if and only if either \(T\equiv 0\) or \(T\) is a \((U,V)\)-operator. This theorem generalizes results by L. B. Beasley [Linear operators which preserve pairs on which the result is additive. J. Korean SIAM, 2, 27-30 (1998)] and A. Guterman [Linear Algebra Appl. 331, 75-87 (2001; Zbl 0985.15018)]. Here a pair \(A,B\in M_{m,n}({\mathbb F})\) is called rank-sum-maximal (rank-sum-minimal) if \(\text{rank}(A+B)= \text{rank}(A)+\text{rank}(B)\) (\(\text{rank} (A+B)=|\text{rank}(A)-\text{rank}(B)|)\). A linear operator \(T\) is a \((U,V)\)-operator if there exist invertible matrices \(U,V\) such that \(T(A)=UAV\) for all \(A\in M_{m,n}({\mathbb F})\) or, if \(m=n\), \(T(A)=UA^tV\) for all \(A\in M_{m}({\mathbb F})\).

MSC:

15A60 Norms of matrices, numerical range, applications of functional analysis to matrix theory
15A45 Miscellaneous inequalities involving matrices
15A03 Vector spaces, linear dependence, rank, lineability

Citations:

Zbl 0985.15018
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References:

[1] Beasley, L.B., Linear operators on matrices: the invariance of commuting pairs, Linear and multilinear algebra, 6, 179-183, (1978) · Zbl 0397.15010
[2] Beasley, L.B., Linear operators which preserve pairs on which the rank is additive, J. Korean SIAM, 2, 27-30, (1998)
[3] Lim, M.H., Linear transformations of tensor spaces preserving decomposable tensors, Publ. inst. math., nouvelle ser., 18, 32, 131-135, (1975) · Zbl 0312.15016
[4] Guterman, A., Linear preservers for matrix inequalities and partial orderings, Linear algebra appl., 331, 75-87, (2001) · Zbl 0985.15018
[5] Watkins, W., Linear maps that preserve commuting pairs of matrices, Linear algebra appl., 14, 29-35, (1976) · Zbl 0329.15005
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