Re: [線代] 幾題eigenvalue問題
※ 引述《willyeh (fen)》之銘言：
: 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滿足 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.
: 麻煩幫解決這二題，謝謝。
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01/01 00:49, , 1F
01/01 00:49, 1F
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