※ 引述《willyeh (fen)》之铭言： : 1 : If P is the matrix thar projects R^n onto a subspace S, explain why every : vector in S is an eigenvector, and so is every vector in complement of S. : What are the eigenvalues?(Note the connection to P^2=P, which means that : eigenvalue^2=eigenvalue) P满足 P^2=P 以及 Im P=S 所以 v \in S => v=Pw for some w in R^n => v=Pw=(P^2)w=P(Pw)=Pv => v is an eigenvector of P with eigenvalue 1. : 2 : (a)Show that the matrix differential equation dX/dt=AX+XB has the solution : X(t)=e^At X(0) e^Bt Let X(t)=e^At X(0) e^Bt. Then X'(t) = (e^At)' X(0) e^Bt + e^At X(0) (e^Bt)' = A(e^At) X(0) e^Bt + e^At X(0) (e^Bt)B =AX(t)+X(t)B and e^At X(0) e^Bt | =X(0) t=0 : (b)Prove that the solutions of dX/dt=AX-XA keep the same eigenvalues for all : time The solution is X(t) = e^At X(0) e^(-At), which is conjugate to X(0) for all t. => X(t) and X(0) have the same set of eigenvalues for all t. : 麻烦帮解决这二题，谢谢。 --

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