(B). (5%) Continued from (A), if the matrix M is given by write down a transformation matrix S that can diagonalize M into D and also write down the corresponding diagonal matrix D.

(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 (λnt) 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.

題組內容

(B). (5%) Continued from (A), if the matrix M is given by
write down a transformation matrix S that can diagonalize M into D and also write down the corresponding diagonal matrix D.

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