Question

Let M be a given symmetric n x n real matrix. let A be a linear operator on real antisymmetric n x n matrices given by AX = MXM
1/ What is the largest possible number of complex eigenvalues that the operator A may have. Justify your answer
2. What is the largest possible number of eigenvalues of A that are not real? Justify your answer
Please, help

Answer 1

given that M is an nxn matrix it follows that the most number of eigenvalues it could have is n, given that M is asymmetrical it follows that either real or complex eigenvalues are possible. Therefore, it seems that n should be the answer to both questions. However, my linear algebra is dusty.

Answer 2

I've got to go on short notice, but it seems like what you're doing is related to this theorem. http://en.wikipedia.org/wiki/Spectral_theorem

If you haven't already, I would recommend reading through it and seeing if you learn anything new.

Answer 3

I don't know what is the role of X there :(

Answer 4

What do you mean by "the role of X". That's a multiplier.

Answer 5

Unfortunately, I also have to log out at the moment, but I think that you should look at the link @KingGeorge left for you.

Answer 6

but A is antisymmetric, so that A^t = -A
M is symmetric, M^t = M, how about X?

Answer 7

I read, but Spectral theorem do nothing here:(

Answer 8

IF you want I'll try to log in later and reply, hopefully someone else can help you out at the moment.

Answer 9

Ok, thanks any way. But when you log in and find out something, please, leave your guidance here.

Answer 10

I'll make an effort to be back later.

Answer 11

thank you