Paul Michael Kielstra

Any function \(f:\mathbb{R}\mapsto\mathbb{R}\) can be written as the sum of an even and an odd function: \(f(x) = \frac{f(x)+f(-x)}{2} + \frac{f(x)-f(-x)}{2}.\) Similarly, any square matrix can be written as the sum of a symmetric and an antisymmetric matrix: \(A = \frac{A + A^t}{2} + \frac{A - A^t}{2}.\) A student of mine recently asked me about this similarity, and, while I described it in lecture as a “standard mathematical trick,” I thought I should sit down and work out the pattern.

So let \(V\) be some vector space, and let \(T\) be a linear operator on \(V\) such that \(T^n = I\) for some \(n\). For instance, we might take \(Tf(x)=f(-x)\) or \(T(A)=A^*\). In both of these cases, \(n=2\). Then the minimal polynomial of \(T\) exists and divides \(p(z)=z^n-1\), which, in complex numbers, splits into distinct linear factors. Therefore \(T\) is diagonalizable, which, even in the infinite-dimensional case, means that \(V\) can be written as a direct sum of the eigenspaces of \(T\).

In the \(n=2\) case, for instance, take once again \(Tf(x)=f(-x)\). Then the two eigenvalues are \(\pm 1\) and the eigenspaces are \(\{f:f(-x)=f(x)\}\) and \(\{f:f(-x)=-f(-x)\}\) respectively. Our formula for writing \(f\) as the sum of an even and an odd function is just a specific special case of the theorem that tells us that we can write the space of real-valued functions as a direct sum of these eigenspaces.

When \(n \geq 3\), the formula is a little more complicated. The eigenvalues are \(e^{2 \pi i k / n}\) for \(k = 1, \dots, n\). Then we can compute that, for any vector \(v\),

\[T\left(\frac{v + e^{2 \pi i k/n}Tv + \cdots + e^{2(n-1)\pi i k/n}T^{n-1}v}{n}\right) \\ = e^{2(n-1)\pi i k/n}\frac{v + e^{2 \pi i k/n}Tv + \cdots + e^{2(n-1)\pi i k/n}T^{n-1}v}{n}.\]

Furthermore,

\[\sum_{k=1}^n \frac{v + e^{2 \pi i k/n}Tv + \cdots + e^{2(n-1)\pi i k/n}T^{n-1}v}{n} = v,\]

since the sum of the \(n\)-th roots of unity is zero. This is therefore our formula for decomposing \(v\) into a sum of components drawn from the eigenspaces of \(T\). It is unique since the sum of eigenspaces is direct.

In the case where \(n=2\), both roots of unity are in fact integers, so the theorem is very easy to apply. For larger \(n\), we need complex numbers, and irrational ones at that. The formulae are less pleasant. That said, it’s still satisfying to be able to apply them. Consider what happens if we take a function \(\mathbb{C}\rightarrow\mathbb{C}\) and rotate the input by \(2\pi/3\). Say that our function is undisturbed if the output is unchanged, slightly annoyed if the output is also rotated by \(2\pi/3\) but is otherwise unchanged, and cheesed off if the output is rotated by \(4\pi/3\) but is otherwise unchanged. Then we can write any complex-valued \(f\) as the sum of one cheesed-off, one slightly-annoyed, and one undisturbed function:

\[\begin{aligned}f(x) &= \frac{f(x) + e^{2 \pi i / 3}f(e^{2 \pi i / 3}x) + e^{4 \pi i / 3}f(e^{4 \pi i / 3}x)}{3} \\ &+ \frac{f(x) + e^{4 \pi i / 3}f(e^{2 \pi i / 3}x) + e^{2 \pi i / 3}f(e^{4 \pi i / 3}x)}{3} \\ &+ \frac{f(x) + f(e^{2 \pi i / 3}x) + f(e^{4 \pi i / 3}x)}{3}.\end{aligned}\]