# I need help for this matrix differential equation!

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• October 16th 2012, 06:15 AM
alphabeta89
I need help for this matrix differential equation!
Let A be a real 3 × 3 matrix, u and v linearly independent vectors in R3 such that Au = u
and Av = v. Suppose w is a vector in R3 such that Aw = w + u + v.

(i) Find all eigenvalues of A.
(ii) Solve the differential system x′ = Ax.

For (ii) I know that 1 is definitely an eigenvalue, is there any other eigenvalues apart from 1? Thanks in advance!
• October 16th 2012, 08:31 AM
FernandoRevilla
Re: I need help for this matrix differential equation!
Quote:

Originally Posted by alphabeta89
Let A be a real 3 × 3 matrix, u and v linearly independent vectors in R3 such that Au = u
and Av = v. Suppose w is a vector in R3 such that Aw = w + u + v.

(i) Find all eigenvalues of A.
(ii) Solve the differential system x′ = Ax.

Prove that $B=\{u,u+u,w\}$ is a basis of $\mathbb{R}^3$. Then, $\begin{Bmatrix}Au=u\\A(u+v)=u+v\\Aw=(u+v)+w\end{ma trix}$ which implies that $J=\begin{bmatrix}{1}&{0}&{0}\\{0}&{1}&{1}\\{0}&{0} &{1}\end{bmatrix}$ is the canonical Jordan form of $A$.
• October 16th 2012, 09:45 AM
HallsofIvy
Re: I need help for this matrix differential equation!
Very nice.
• October 16th 2012, 10:10 AM
FernandoRevilla
Re: I need help for this matrix differential equation!
Quote:

Originally Posted by HallsofIvy
Very nice.

Nice to hear that. :)