It is evident that if two of these three symmetries are present, then the third is also present, due to the relation formula_43. The aforementioned discrete symmetries label 10 distinct discrete symmetry classes, which coincide with the Altland–Zirnbauer classes of random matrices. A bulk Hamiltonian in a particular symmetry group is restricted to be a Hermitian matrix with no zero-energy eigenvalues (i.e. so that the spectrum is "gapped" and the system is a bulk insulator) satisfying the symmetry constraints of the group. In the case of formula_44 dimensions, this Hamiltonian is a continuous function formula_38 of the formula_46 parameters in the Bloch momentum vector formula_47 in the Brillouin zone; then the symmetry constraints must hold for all formula_48. Given two Hamiltonians formula_49 and formula_50, it may be possible to continuously deform formula_49 into formula_50 while maintaining the symmetry constraint and gap (that is, there exists continuous function formula_53 such that for all formula_54 the Hamiltonian has no zero eigenvalue and symmetry condition is maintained, and formula_55 and formula_56). Then we say that formula_49 and formula_50 are equivalent. However, it may also turn out that there is no such continuous deformation. in this case, physically if two materials with bulk Hamiltonians formula_49 and formula_50 respectively neighbour each other with an edge between them, when one continuously moves across the edge one must encounter a zero eigenvalue (as there is no continuous transformation that avoids this). This may manifest as a gapless zero energy edge mode or an electric current that only flows along the edge. An interesting question is to ask, given a symmetry class and a dimension of the Brillouin zone, what are all the equivalence classes of Hamiltonians. Each equivalence class can be labeled by a topological invariant; two Hamiltonians whose topological invariant are different cannot be deformed into each other and belong to different equivalence classes. For each of the symmetry classes, the question can be simplified by deforming the Hamiltonian into a "projective" Hamiltonian, and considering the symmetric space in which such Hamiltonians live. These classifying spaces are shown for each symmetry class:  For example, a (real symmetric) Hamiltonian in symmetry class AI can have its formula_61 positive eigenvalues deformed to +1 and its formula_62 negative eigenvalues deformed to -1; the resulting such matrices are described by the union of real Grassmannians formula_63  The strong topological invariants of a many-band system in formula_46 dimensions can be labeled by the elements of the formula_46-th homotopy group of the symmetric space.