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As far as its coordinate representation is concerned, the domain of a linear transformation will eventually (i.e. after infinitely many iterations of the transformation) be mapped onto the dominant eigenspace.

True or False?

p.s. Revised hypothesis: Let T be a linear transformation that has a square matrix representation with no complex eigenvalues. A nonzero vector transformed iteratively by T tends towards the dominant eigenspace (oscillatingly if the dominant eigenvalue has a negative sign). In coordinate space, T will, after infinitely many iterations, evetually map the original domain onto the dominant eigenspace.

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    @Gerry Sorry not yet, because I think I was skipping ahead of myself (my conjecture was based on very elementary and simplified material from old coursework), and realised that I can better deal with my nonsensical understanding once I acquire the appropriate tools. I will come back to this topic in a few days after I have read up on applications of eigenvalues to dynamical systems incl. long-term behaviour. Thanks I really really appreciate your time. You rock.2012-08-07

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Call the linear transformation $T$, and the vector space on which it acts, $V$. Assume $V$ is finite-dimensional.

In the special case where the $V$ has a basis $B$ consisting of eigenvectors of $T$, and there is a dominant eigenvalue $\lambda$, the following is true: if $v$ is any element of $V$, and if the expression for $v$ with respect to the basis $B$ has a nonzero component in the eigenspace $W$ of $\lambda$, then $\lim_{n\to\infty}\lambda^{-n}T^nv$ exists and is nonzero and is in $W$.

The easiest way to see this is to write $v$ as a linear combination of vectors in $B$ and then compute $\lim_{n\to\infty}\lambda^{-n}T^nv$ (this will also give you some idea what to do when the conditions are not met). That is, let $B=\{{x_1,x_2,\dots,x_r\}}$ and let $v=a_1x_1+a_2x_2+\cdots+a_rx_r$ where $Tx_i=\lambda_ix_i$. Then $T^nv=a_1\lambda_1^nx_1+a_2\lambda_2^nx_2+\cdots+a_r\lambda_r^nx_r$ Now suppose $\lambda_1$ is dominant, that is, $|\lambda_1|\gt|\lambda_i|$ for $i=2,3,\dots,r$. Then $\lambda_1^{-n}T^nv=a_1x_1+a_2(\lambda_2/\lambda_1)^nx_2+\cdots+a_r(\lambda_r/\lambda_1)^nx_r$ and when you take the limit as $n\to\infty$ you get $a_1x_1$.

Things get more complicated if there isn't a single dominant eigenvalue but you can still use this approach and get pretty far with it.

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    I've read your answer and also a bit on eigens and discrete dynamical systems, and it seems that my conjecture/generalisation was missing quite a few qualifications, although the heart was in the right place haha. There cannot be complex eigenvalues, the component of the initial vector in the direction(s) of the dominant eigenspace must be nonzero, and apparently the dominant eigenvalue must be the only one that is at least 1 (this last qualification puzzles me though).... Thanks for your help.2012-08-09
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What about a 2D-rotation by an irrational angle? Will this ever (after iterating) have an eigenspace in $\mathbb R^2$.

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    Ok I have read up more and just learnt that all matrices with complex eigenvalues have rotational information encoded within. Fascinating!2012-08-08