[ ] Let be any matrices with entries from and Let and denote the trace of and the identity matrix respectively. Find some such that
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i don't have to mention that i'm not interested in a direct solution. anyway, here is a hint for those who like the problem: Spoiler: a clever use of Cayley-Hamilton theorem will solve the problem very quickly! by the way, the result can be extended to commuting matrices but the solution to this general case is much harder.
This is a particular consequence of a very well-known result. For any matrices and In particular, if then Spoiler: where and . Incidentally, holds for all matrices and . What is ?
Last edited by halbard; Aug 20th 2010 at 01:12 AM.
Originally Posted by halbard This is a particular consequence of a very well-known result. For any matrices and well, assuming that the above holds, your solution would obviously be correct. my solution is different from yours. Incidentally, holds for all matrices and . What is ? by Cayley-Hamilton and the fact that
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