I know that Sylvester's Law tells us that the number of positive (resp. negative, null) eigenvalues of a real symmetric matrix is invariant under diagonalization.
I also believe that this can be alternately formulated as the dimension of the maximal positive-definite (resp. negative definite) subspaces being invariant.
I belive I understand each statement, but how can I show that these are equivalent? I believe I'm missing a crucial detail concerning the relationship between eigenvalues and the dimension of subspaces.
I suppose this question could be: how can I prove that the number of positive eigenvalues of a space is equal to the dimension of a maximal positive-definite subspace?