For example, a (real symmetric) Hamiltonian in symmetry class AI can have its n positive eigenvalues deformed to +1 and its N − − n negative eigenvalues deformed to -1; the resulting such matrices are described by the union of real Grassmannians ⋃ ⋃ n = 0 ∞ ∞ G r ( n , N ) = ⋃ ⋃ n = 0 ∞ ∞ O ( N ) / O ( n ) × × O ( N − − n )