  In this paper we generalize the notion of the comparative index for the pair
of Lagrangian subspaces which has fundamental applications in oscillation
theory of symplectic difference systems and linear differential Hamiltonian
systems. We introduce cyclic sums $\mu_c^{\pm}(Y_1,Y_2,\dots,Y_m),\,m\ge 2$ of
the comparative indices for the set of $n-$ dimensional Lagrangian subspaces.
We formulate and prove main properties of the cyclic sums, in particular, we
state connections of the cyclic sums with the Kashiwara index. The main results
of the paper connect the cyclic sums of the comparative indices with the number
of positive and negative eigenvalues of $mn\times mn$ symmetric matrices
defined in terms of the Wronskians $Y_i^T \,J\, Y_j,$ $i,j=1,\dots,m.$ We also
present first applications of the cyclic sums of the comparative indices in the
oscillation theory of the discrete symplectic systems connecting the number of
focal points of their principal solutions with the negative and positive
inertia of symmetric matrices.
