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Quadratic Transformations on Matrices: Rank Preservers

Authors
Journal
Journal of Algebra
0021-8693
Publisher
Elsevier
Publication Date
Volume
179
Issue
2
Identifiers
DOI: 10.1006/jabr.1996.0024
Disciplines
  • Mathematics

Abstract

Abstract Let Fbe an algebraically closed field of characteristic not 2, and let X=( X i j ) be the n× nmatrix whose entries X i j are independent indeterminates over F. Now let Q( X)=( q i j ( X)) be another n× nmatrix each of whose entries q i j ( X) is a quadratic F-polynomial in the X i j . The main result in this paper is: for n≥5, Q( X) satisfies rank( A 2)= rimplies rank( Q( A))= r, for all A∈ F n× n for r=0, 1, and 2, if and only if there exist invertible matrices P 1, P 2in F n× n such that either Q( X)= P 1 X 2 P 2or Q( X)= P 1( X 2) t P 2.

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