Linear transformations which preserve fixed rank |
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Authors: | LeRoy B Beasley |
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Institution: | 1712 Dearborn Street Caldwell, Idaho, 83605, USA |
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Abstract: | Let Mm, n(F) denote the set of all m×n matrices over the algebraically closed field F. Let T denote a linear transformation, T:Mm, n(F)→Mm, n(F). Theorem: If max(m, n)?2k?2, k?1, and T preserves rank k matrices i.e.?(A)=k implies ?(T(A))=k], then there exist nonsingular m×m and n×n matrices U and V respectively such that either (i) T:A→UAV for all A?Mm, n(F), or (ii) m=n and T:A→UAtV for all A?Mn(F), where At denotes the transpose of A. |
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