Your question is how to get expm(c*A) given A for a scalar c.
Diagonalize A such that A*V = V*D.
It follows that (c*A)*V = V*(c*D) and - if V is invertible - c*A = V*(c*D)*inv(V).
Thus expm(c*A) = V*diag(exp(diag(c*D)))*inv(V) .
Summarizing: Saving V and inv(V) (if V is invertible) gives you expm(c*A) from a diagonalization of the matrix A as A = V*D*inv(V).
A = [1 2; 3 4];
[V,D] = eig(A)
expm(A)
V*diag(exp(diag(D)))*inv(V)
c = 2;
expm(c*A)
V*diag(exp(diag(c*D)))*inv(V)
