Sylvester's law of inertia

Sylvester's law of inertia says that when you diagonalize a real quadratic form into a signed sum of squares, the NUMBER of positive coefficients, the number of negative coefficients, and the number of zeros are the same no matter which diagonalizing change of variables you used. The individual coefficients are not pinned down — you can rescale them — but their signs, counted up, are an iron invariant.

Stated for matrices: if a real symmetric matrix B is congruent to two diagonal matrices, those two diagonals have the same count of positive, negative, and zero entries. So congruence can move the diagonal values around freely, but it can never convert a plus into a minus or create or destroy a zero. That triple of counts is what congruence truly remembers.

The proof is a clean dimension-counting argument. Suppose the form were positive on some p-dimensional subspace and a competing diagonalization gave p' positives with p' < p; the positive subspace of dimension p and the nonpositive subspace of dimension n - p' would have dimensions summing past n, forcing a nonzero common vector that is both positive and nonpositive — a contradiction. So p = p', and likewise for the negatives.

This is the theorem that makes 'signature' a legitimate concept rather than an artifact of bookkeeping. Without inertia, two people completing the square in different orders might report different answers; inertia guarantees they agree on the only thing that matters. It is the linear-algebra ancestor of the signature of a metric in relativity, where the (+,-,-,-) splitting is physically meaningful precisely because it cannot be changed by coordinates.