trivial-observations

[trivial-observations]

Idempotent Linear Maps

%view at https://trivial-observations.tumblr.com/post/734896076331646976/idempotent-linear-maps to render the math

Just some nice facts about linear maps when they are idempotent. These are from the last two exercises from chapter 9 of Napkin.

Lemma 1. Let \(P \colon V \longrightarrow V\) be a linear map, where \(V\) is a (not necessarily finite-dimensional) vector space. Suppose that \(P\) is idempotent, i.e. \(P(P(v)) = P(v)\), or equivalently that \(P\) is the identity on its image. Then,

\[V = \text{ker } P \oplus \text{im } P.\]

Thus, we can think of \(P\) as projection onto the subspace \(\text{im } P\).

Proof. Let \(u \in V\). Suppose \(w \in \text{ker } P\) such that \(u = v + w\). This forces \(P(u) = P(v)\), and since \(P\) is idempotent \(v \in \text{im} V\). \( \blacksquare\)

This is fun because linear endomorphisms of finite-dimensional vector spaces are idempotent when raised to a power.

Lemma 2. Let \(V\) be a finite-dimensional vector space, and let \( T \colon V \longrightarrow V \) be a linear map. Let \(T^n\) denote \(T\) composed with itself \(n\) times. Then there is some \(N\) for which \(T^N\) is idempotent.

Proof. Notice that for each \(k\in \textbf{N}\), \(\text{im } T^{k+1} \subseteq \text{im }T^k\). Hence, \(\text{dim } \text{im } T^{k+1} \leq \text{dim } \text{im } T^k\). Since \(V\) is finite-dimensional, there is some \(m\) such that \(\text{dim } \text{im } T^m\) is minimal.

Now, the restriction of \(T^{m^j}\) on \(\text{im } T^m\), where \(j \in \mathbf{N}\), is an isomorphism. Thus, for any basis of \(\text{im } T^m\), it gets permuted by \(T^{m^j}\), implying the existence of some \(j\) for which \(T^{m^j}\) is the identity on \(\text{im } T^m\), and \(N = m^j\) works. \( \blacksquare \)

Combining this result with the previous one, we get

Theorem 3. Let \(V\) be a vector space and \(T \colon V \longrightarrow V\) a linear map. Then there is an \(N \in \textbf{N}\) such that

\[V = \text{im } T^N \oplus \text{ker } T^N.\]