Technique to find the CFL condition using the Galerkin method in space and finite-difference in time?

I am using the Galerkin method (Discontinuous to be precise) to discretize in space the scalar linear wave equation and the explicit second order centered finite difference scheme to discretize in time, resulting in a semi-discrete system of the kind:

$Mddot{U} + KU=0, quad U(0)=U_0, quad dot{U}(0) = U_1,$

where $M, K$ are the mass and stiffness matrix of the Galerkin method. Now the Idea to study the stability is to observe that if the bilinear form $a$ from which $K$ originates has an $L^2$-orthonormal basis of eigen-functions on the discrete space $V_h$: $a(w_i, v)=lambda_i (w_i, v), quad forall v in V_h, quad i = 1 dots N, quad dim{V_h} =N,$

then after using the time integration scheme and expressing $u^k= sum_{j=1}^N u_j^kw_j$:

$(u^{n+1}-2u^n+u^{n-1}, w_i) + Delta t^2a(u^n,w_i)=0$

becomes, using the fact that $(w_j, w_i) = delta_{i,j}:$

$u^{n+1}_i = (2-lambda_i Delta t ^2)u_i^n – u_i^{n-1}, quad forall i = 1 dots N$

Now the point is basically solving this second order difference equation and asking that it doesn’t oscillate or explode and this results in the CFL condition of the kind $Delta t le C h$ (after bounding the $lambda_i$ with the mesh size $h$).

So being a second order difference equation there can be cases where the discriminat is positive, negative (two complex roots) or zero. Is this the right track to do these kind of stability analysis? Are these requirements all that is needed to have the CFL condition? Thanks a lot for clarifying.