Simultaneous congruence of convex compact sets of Hermitian matrices with constant rank |
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Institution: | Oregon State University Corvallis, Oregon 97331, U.S.A.;University of Toronto Toronto, Ontario, Canada;Department of Mathematics San Diego State University San Diego, California 92182, U.S.A.;Tel-Aviv University Tel-Aviv, Israel;Arizona State University Tempe, Arizona 85287, U.S.A. |
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Abstract: | Let S be a compact convex set of n × n hermitian matrices (n ⩾ 2). Suppose every member of S is nonsingular and has exactly one negative eigenvalue. Let (ε1,…,εn) be any ordered n-tuple from the set {- 1, 1}. One of our main results is that a nonsingular matrix X exists such that, for every A in S and every 1 ⩽ j ⩽ n, the (j, j) entry of X1AX has sign εj. A similar result, with only negative εj allowed, is proved also for a compact convex set S of n × n hermitian matrices such that every member of S has the same rank and exactly one negative eigenvalue. |
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