Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Let $V$ be the complex vector space given by $$V = \{a\sin(x) + b\cos(x) +cx\sin(x) + dx\cos(x)\ \; | \; a, b, c, d \in \mathbb{C}\}$$

Let $D: V \rightarrow V$ be the linear map given by $D(f) = f'$.

I need to work out the characteristic equation and complex eigenvalues of D and $m_D x$.

I thought the first step would be to calculate the derivative but I don't know where to go from here?

share|improve this question

1 Answer 1

Well, you'll have to find a matrix representing the tranformation, won't you.

You have that $\sin(x),\cos(x), x\sin(x), x\cos(x)$ form a basis for your vector space. I'll call these $a,b,c,d$ respectively. Observe the effects of $D$ on the basis: $$a\mapsto b\\ b\mapsto -a \\ c\mapsto a+d\\ d\mapsto b-c$$

Find the matrix in terms of this basis, and you can begin to find eigenvalues.

share|improve this answer
    
I thought that it had to be a square matrix to get a characteristic equation? I don't know how to make this into a square matrix? –  user1131194 Nov 1 '12 at 0:11
    
@user1131194 This is important for you to spend time thinking about so that you can understand what's going on. To get you started, the coordinates for $a$ are $[1,0,0,0]$, the coordinates of $b$ are $[0,1,0,0]$ etc for $c$ and $d$. Then you have to work out a matrix such that if you multiply these coordinate vectors, you get the results predicted above. –  rschwieb Nov 1 '12 at 1:05

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.