A Sledgehammer Proof of the Spectral Theorem for Normal Matrices

My research is about the structure of homomorphisms between certain C*-algebras. To illustrate what I mean and its possible importance, consider an elementary case given below.

Proposition. Let X be a compact, Hausdorff space, N be a positive integer, and \phi\colon C(X) \to M_N be a homomorphism. There exist mutually orthogonal rank one projections p_n  and points \xi_n\in X for n=1,2,\dotsc,N such that

\phi(f) = \sum_{n=1}^{N} f(\xi_n) p_n

for all f \in C(X).

Proof: Since C(X)/\ker\,\phi is a commutative C*-algebra, by Gelfand’s representation theorem, there exists a compact Hausdorff space Y such that C(X)/\ker\,\phi \cong C(Y). Furthermore, since there is a surjective homomorphism from C(X) onto C(Y), there is a continuous embedding from Y into X. We assume, without loss of generality, that Y\subseteq X and that the canonical projection homomorphism is given by restriction.

Let \phi'\colon C(Y) \to M_N be the injective homomorphism induced by \phi. Since \phi' is an injective linear map and M_N is finite dimensional, Y is finite (i.e. consists of finitely many points). Label these points \xi_n and let \chi_n \in C(Y) denote the indicator functions for \{\xi_n\}. Notice that \phi'(\chi_n) are mutually orthogonal projections. Each projection \phi'(\chi_n) can be decomposed into the sum of mutually orthogonal rank one projections p_n. By relabeling and repeating \xi_n as necessary, we assume that p_n is associated with the corresponding point \xi_n.

Let \pi_n\colon M_N \to p_n M_N p_n be the usual positive linear map. Since p_n has rank one, p_n M_N p_n \cong \mathbb{C}. So it is easy to see that \pi_n \circ \phi' is a state of C(Y) and so \pi_n \circ \phi'(f) = f(\xi_n). Therefore,

\phi(f) = \sum_{n=1}^{N}\pi_n\circ \phi(f) = \sum_{n=1}^{N} f(\xi_n) p_n.    \Box

We can rephrase this proposition to resemble linear algebra more. Since rank one projections correspond to one-dimensional subspaces, we can choose an orthonormal basis (\delta_n), where \delta_n is a unit vector in the range of p_n. So there exists a unitary u \in M_N such that u\delta_n = \epsilon_n, where \epsilon_n is the standard basis for \mathbb{C}^n. So our proposition above can be restated as:

Proposition. Let X be a compact, Hausdorff space, N be a positive integer, and \phi\colon C(X) \to M_N be a homomorphism. There exist points \xi_n\in X for n=1,2,\dotsc,N and a unitary matrix u\in M_N such that

u\phi(f)u^{*} = \left(\begin{array}{c c c c}f(\xi_1) & 0 & \cdots & 0 \\ 0 & f(\xi_2) & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & 0 & f(\xi_N) \end{array}\right)

for all f \in C(X).

To apply this theorem, let a \in M_N be a normal matrix. Then by continuous functional calculus, there is a homomorphism from C(X) to M_N denoted f \mapsto f(a), where X = \mathrm{spec}(a). In particular, the inclusion function z \mapsto z is sent to the matrix a. By applying the proposition to this homomorphism and plugging in the inclusion function, we obtain:

Theorem (Spectral Theorem for Normal Matrices). Let N be a positive integer. For any normal matrix a \in M_N, there exist \xi_n\in \mathbb{C} for n=1,2,\dotsc,N and a unitary matrix u\in M_N such that

u a u^{*} = \left(\begin{array}{c c c c}\xi_1 & 0 & \cdots & 0 \\ 0 & \xi_2 & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & 0 & \xi_N \end{array}\right)

As stated, this is the rather significant finite dimensional spectral theorem using rather excessively powerful tools for the proof. But we do obtain a stronger version for “free”. Let S\subseteq M_N be a set of commuting normal matrices. Using the spectral theorem above, C^{*}(S) is a commutative C*-subalgebra of M_N and so by Gelfand’s representation theorem, C^{*}(S) \cong C(X) for some compact Hausdorff space X. So we obtain a homomorphism from C(X) to M_N defined by composing the Gelfand transform with inclusion. Since each element of S, corresponds to some continuous function of X, by applying our proposition and plugging those functions, we obtain a simultaneous diagonalization of the matrices in S:

Theorem. Let N be a positive integer. For any set S \subseteq M_N of commuting normal matrices, there exists a unitary u \in M_N, such that for any a\in S, there exist \xi_n\in \mathbb{C} for n=1,2,\dotsc,N such that

u a u^{*} = \left(\begin{array}{c c c c}\xi_1 & 0 & \cdots & 0 \\ 0 & \xi_2 & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & 0 & \xi_N \end{array}\right)