  The eigenvalues of a parameter-dependent Hamiltonian matrix form a band
structure in parameter space. In such $N$-band systems, the quantum geometric
tensor (QGT), consisting of the Berry curvature and quantum metric tensors, is
usually computed from numerically obtained energy eigenstates. Here, an
alternative approach to the QGT based on eigenprojectors and (generalized)
Bloch vectors is exposed. It offers more analytical insight than the eigenstate
approach. In particular, the full QGT of each band can be obtained without
computing eigenstates, using only the Hamiltonian matrix and the respective
band energy. Most saliently, the well-known two-band formula for the Berry
curvature in terms of the Hamiltonian vector is generalized to arbitrary $N$.
The formalism is illustrated using three- and four-band multifold fermion
models that have very different geometrical and topological properties despite
an identical band structure. From a broader perspective, the methodology used
in this work can be applied to compute any physical quantity or to study the
quantum dynamics of any observable without the explicit construction of energy
eigenstates.
