I am given $W$ a subspace of real $n$-dimensional matrices which are symmetric and pairwise commuting. I have to prove that $dim(W) \leq n$.
I have read some facts about commuting matrices over an algebraic closed field, but these do not apply in this case.
As hinted in the above comment, we must use the fact that the matrices must be simultaneously (orthogonally) diagonalizable.
That is, all matrices in the set have the form $$ U\pmatrix{\lambda_1\\ &\ddots\\ && \lambda_n} U^T $$ for some orthogonal $U$. The space of such matrices (for a fixed $U$) is a subspace of $\Bbb R^{n \times n}$ with dimension $n$.
