  We generalize the method of computing functional determinants with a single
excluded zero eigenvalue developed by McKane and Tarlie to differential
operators with multiple zero eigenvalues. We derive general formulas for such
functional determinants of $r\times r$ matrix second order differential
operators $O$ with $0 < n \leqslant 2r$ linearly independent zero modes. We
separately discuss the cases of the homogeneous Dirichlet boundary conditions,
when the number of zero modes cannot exceed $r$, and the case of twisted
boundary conditions, including the periodic and anti-periodic ones, when the
number of zero modes is bounded above by $2r$. In all cases the determinants
with excluded zero eigenvalues can be expressed only in terms of the $n$ zero
modes and other $r-n$ or $2r-n$ (depending on the boundary conditions)
solutions of the homogeneous equation $O h=0$, in the spirit of Gel'fand-Yaglom
approach. In instanton calculations, the contribution of the zero modes is
taken into account by introducing the so-called collective coordinates. We show
that there is a remarkable cancellation of a factor (involving scalar products
of zero modes) between the Jacobian of the transformation to the collective
coordinates and the functional fluctuation determinant with excluded zero
eigenvalues. This cancellation drastically simplifies instanton calculations
when one uses our formulas.
