  Even index pairings are integer-valued homotopy invariants combining an even
Fredholm module with a $K_0$-class specified by a projection. Numerous
classical examples are known from differential and non-commutative geometry and
physics. Here it is shown how to construct a finite dimensional selfadjoint and
invertible matrix, called the spectral localizer, such that half its signature
is equal to the even index pairing. This makes the invariant numerically
accessible. The index-theoretic proof heavily uses fuzzy spheres.
