How to eliminate scale factors on a pair of matrices....
From: Vincent Carla (vincent.carla_at_free.fr)
Date: 08/16/04
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Date: Mon, 16 Aug 2004 15:44:03 +0000 (UTC)
Hello,
I'm stuck with the following problem that I can't solve.
I consider two 3-by-3 (real) matrices defined as
B1 = s1*H*A1*H^T
B2 = s2*H*A2*H^T
where s1 and s2 are non zero real numbers,H is a full-rank 3-by-3 (real) matrix (H^T denotes its transpose) and A1 and A2 are two 3-by-3 (real) matrices such that rank(A1)=3 and rank(A2)<=3.
I'm seeking some assumptions on B1, B2 that will ensure that
s2/s1 equals a known quantity ie, only depending on A1 and A2.
When rank(A2) = 3, it suffices to assume that
det(B1) = det(B2) = 1
so
det(inv(B2))*det(B1) = (s1/s2)^3*det(A1)/det(A2) = 1
yields
(s1/s2)^3 = det(A2)/det(A1)
When rank(A2)<=3, I could not think of any assumption on B1, B2 that will help me
to solve the problem in a similar way.
Thanks a lot for any suggestion.
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