Deducing Properties of M^2=I Given an nxn Matrix

[Ultraworld]

Given an n by n matrix M with complex coefficients such that M² = I and M is not equal to I.

What can I deduce from it. e.g. what does it say about the eigenvalues?



edit: of course M ^-1 = M.

 

[matt grime]

M satisfies the polynomial x^2-1. M does not satisfy x-1. That tells you that the minimal poly is either x^2-1 or x+1. That tells you all of the eigenvalues.

 

[Ultraworld]

he that looks good thanks.

Im going to figure it out.

http://en.wikipedia.org/wiki/Minimal_polynomial_(linear_algebra)
http://mathworld.wolfram.com/MatrixMinimalPolynomial.html

 

[Ultraworld]

So -1 is an eigenvalue of M.

However am I guaranteed that the equation M x = - x has a non-trivial solution?

EDIT: I think the answer is yes cause the dimension of the Eigenspace is the same as the geometric multiplicity of -1 http://en.wikipedia.org/wiki/Eigenvalue (see definitions)

EDIT2: I am sure the answer is yes

 

[matt grime]

If -1 is not an eigen value of M, then all its eigenvalues are 1, and it must satisfy x-1, so M would be the identity matrix, which we are told it is explicitly not.

By definition, if t is an eigenvalue of M, then there is an eigenvector with eigenvalue t.

 

[Ultraworld]

thanks Matt. This was part of a bigger problem and that bigger one is now solved.

 

[Ultraworld]

I can make my solution much easier.

Given a n by n matrix N with complex coefficients and det N != 0. Does N has a non-zero eigenvalue?

 

[matt grime]

Yes. Obviously.

 

[HallsofIvy]

Any determinant has at least one eigenvalue (in the complex numbers) because every polynomial has at least one complex solution. Since the determinant is not 0, that eigenvalue is not 0, therefore ---