Sylvester’s Law of Inertia

For a real symmetric matrix, inertia is the ordered triple of positive, negative and zero eigenvalue counts. A congruence A = S^TDS with invertible S preserves those counts. It need not preserve the eigenvalues themselves, unlike similarity.

Reproduce the coordinate change

Open Sylvester’s Law of Inertia and compare the matrices and quadratic-form cards.

1. Click Reset. The controls d1 = 1, d2 = −1, S11 = 1, S12 = 1, S21 = 0, S22 = 1 and x1 = x2 = 1 reproduce the worked example: A inertia (1,1,0), both forms 3.
2. Set d1 to 0 and d2 to −2. Set S11 to 2 and S21 to 1, leaving S12 = S22 = 1. A becomes [[−2,−2],[−2,−2]], with eigenvalues −4 and 0 and inertia (0,1,1).
3. Set x1 to 0 and x2 to 2. Sx = [2,2] and both quadratic forms equal −8. Changing x tests equality without changing inertia.
4. Set S11 to 1, S12 to 2, S21 to 1 and S22 to 2. The rows coincide and det S = 0; the app stops asserting inertia preservation.
5. Click Reset, then set d1 = d2 = 0. A is the zero matrix, inertia is (0,0,2), and both forms vanish.