Consider the following equivalence relation on the set of nxn matrices.
X is equivalent to Y if there is an invertible matrix A so that Y=AXA^t, where A^t is the transpose of A. How can I describe the set of equivalence classes? For instance, what can I take as representatives, or what are invariants?
I understand the problem when X is symmetric or antisymmetric, it is the general case that I'm wondering about.
2006-09-21 11:28:24 · 3 answers · asked by Steven S 3 in Science & Mathematics ➔ Mathematics
A can be any invertible matrix, not just unitary.
2006-09-21 11:53:11 · update #1
if X is the identity, what kind of Y's do you get?
Y=AA^t, this is set would give you a model for all other equivalence classes.
for example,
Y is symmetric, since Y=Y^t, they also have positive determinant:
det Y = (detA)^2
2006-09-25 02:37:36 · answer #1 · answered by Anonymous · 0⤊ 0⤋
Interesting question.
You can show for example that equivalent matrices have the same rank and they can be obtained from each other by a sequence of elementary row and column operations.
2006-09-21 11:39:04 · answer #2 · answered by jarynth 2 · 0⤊ 1⤋
See references below.
2006-09-28 12:06:30 · answer #3 · answered by frank 7 · 0⤊ 1⤋