Factor vector out of a matrix

Question

I am wondering how to factor a[1],a[2],...,a[4] out of sim so that I can write sim as the form of a matrix times column vector a. (a[1],a[2],...,a[4] are the component of vector a) ?

My ultimate goal is to compute the Omega when the determinant of the matrix is zero.

Nmax = 4;
x[0][t] = 0;
x[Nmax + 1][t] = 0;

T = 1/2 Sum[m[i] x[i]'[t]^2, {i, 1, Nmax}];
U = 1/2 Sum[k[i] (x[i][t] - x[i - 1][t])^2, {i, 1, Nmax + 1}];

L = T - U;

EL[q_] := D[L, q] - D[D[L, D[q, t]], t]

eigen = Table[EL[x[i][t]], {i, 1, Nmax}] // MatrixForm;

x[i_][t_] = a[i] E^(I \[Omega] t);

sim = eigen /. t -> 0

Answer 1

Because of the linear relationship, you can simply get the desired matrix by calculating the Jacobian, as follows:

matrix = D[sim, {Array[a, Nmax]}];

The Jacobi matrix is by definition the linear approximation to a (differentiable) multivariate function. Here, the variables a[1]...a[4] are therefore used as the independent variables. It's important to have these variables as a list of the form {{a[1]... a[4]}} (two List levels) to get the derivatives arranged in the form of a matrix the way it has to be for the Jacobian. See also the question How to make Jacobian automatically in Mathematica.

What follows is just the verification that the result is correct:

Simplify[matrix.Array[a, Nmax] == sim]

(* ==> True *)

MatrixForm[matrix]