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November 30

Graph Ramsey Theory

I wish to study Graph Ramsey Theory on my own. What books or material should I read to develop a thorough grasp of this. All my searches have yielded books on Graph Theory which have at most a section (not even a chapter) dedicated to Graph Ramsey theory. Can someone name any good resources from where I can learn independently. Thanks-Shahab (talk) 09:36, 30 November 2010 (UTC)[reply]

Symmetric Matrix

I have to show, given a symmetric matrix S such that S raised to the power k is the identity, k greater than or equal to one, that S squared is necessarily the identity. I haven't got a clue where to start with this one. Can someone give me some help? Thanks 131.111.163.135 (talk) 14:07, 30 November 2010 (UTC)[reply]

I assume you are talking about real-valued matrices, as otherwise it's not even true. Then the property follows easily from the fact that real symmetric matrices are diagonalizable.—Emil J. 14:16, 30 November 2010 (UTC)[reply]

You're right, I forgot to mention that it's a real matrix. So I just need some non-singular matrix P st

P^{T}SP=D

where D is a diagonal matrix then? And I can pick a P st D has entries plus or minus one on the diagonal? Thanks. 131.111.163.135 (talk) 14:28, 30 November 2010 (UTC)[reply]

You actually need P⁻¹SP = D. (You can in fact pick P so that P⁻¹ = P^T, but that's irrelevant here.) Then D^k = P⁻¹S^kP = I, which forces D to have ±1 on the diagonal (as there are no other roots of unity in R).—Emil J. 15:07, 30 November 2010 (UTC)[reply]