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Given that I have a matrix of second order differential equations of this form: alt text

Where M, x, C, K are matrix and vectors.

I can decomposed the solutions into different eigenvalues and eigenvectors, as dictacted by the theory of eigenvalue problem, and then solve the equations for each mode of eigenvectors, provided that I have the initial condition for the x and the first derivative of x.

My question is, if the initial conditions are unknown, is there anyway I can still tell the relative magnitude for different eigenvectors?

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No. Your equations form a linear inhomogeneous system of ODEs, so any linear combination of the eigenvectors (of the homogeneous problem) could be added to any "particular solution" to yield a solution for some other choice of initial conditions. Thus you could only get a condition on relative magnitudes if you had some restrictions on your initial conditions.

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