題組內容

(A). (5%) Consider a system of linear differential equations in the following vector form:
Here v(t) is an n-dimensional column vector and M is an n x n matrix. Assume that the matrix M can be diagonalized by a similarity transformation:

Here S is the transformation matrix and D is a diagonal matrix with λ₁,λ₂,... ,λn as its diagonal elements. The initial values of the equations form the vector v(0) and P(t) is a diagonal matrix with exp (λ₁t),exp(λ₂t), ,exp (λ_nt) as its diagonal elements. Please express the solution v(t) for t＞ 0 in terms of v(0), S and P(t) and explain why.