Linear Algebra/Self-Composition

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A linear transformations $t:V\to V$, because it has the same domain and codomain, can be iterated. That is, compositions of $t$ with itself such as $t^2=t\circ t$ and $t^3=t\circ t\circ t$ are defined.

Note that this power notation for the linear transformation functions dovetails with the notation that we've used earlier for their squared matrix representations because if ${\mathrm{R}\mathrm{e}\mathrm{p}}_{B,B}(t)=T$ then ${\mathrm{R}\mathrm{e}\mathrm{p}}_{B,B}(t^j)=T^j$.

These examples suggest that on iteration more and more zeros appear until there is a settling down. The next result makes this precise.

This graph illustrates Lemma 1.3. The horizontal axis gives the power $j$ of a transformation. The vertical axis gives the dimension of the rangespace of $t^j$ as the distance above zero— and thus also shows the dimension of the nullspace as the distance below the gray horizontal line, because the two add to the dimension $n$ of the domain.

As sketched, on iteration the rank falls and with it the nullity grows until the two reach a steady state. This state must be reached by the $n$-th iterate. The steady state's distance above zero is the dimension of the generalized rangespace and its distance below $n$ is the dimension of the generalized nullspace.

Exercises

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