# A Sledgehammer Proof of the Spectral Theorem for Normal Matrices

2013/09/04 2 Comments

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.

Let be a compact, Hausdorff space, be a positive integer, and be a homomorphism. There exist mutually orthogonal rank one projections and points for such thatProposition.for all .

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

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

Let be the usual positive linear map. Since has rank one, . So it is easy to see that is a state of and so . Therefore,

.

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 , where is a unit vector in the range of . So there exists a unitary such that , where is the standard basis for . So our proposition above can be restated as:

PropositionLet be a compact, Hausdorff space, be a positive integer, and be a homomorphism. There exist points for and a unitary matrix such that.for all .

To apply this theorem, let be a normal matrix. Then by continuous functional calculus, there is a homomorphism from to denoted , where . In particular, the inclusion function is sent to the matrix . By applying the proposition to this homomorphism and plugging in the inclusion function, we obtain:

Theorem (Spectral Theorem for Normal Matrices)Let be a positive integer. For any normal matrix , there exist for and a unitary matrix such that.

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 be a set of commuting normal matrices. Using the spectral theorem above, is a commutative C*-subalgebra of and so by Gelfand’s representation theorem, for some compact Hausdorff space . So we obtain a homomorphism from to defined by composing the Gelfand transform with inclusion. Since each element of , corresponds to some continuous function of , by applying our proposition and plugging those functions, we obtain a simultaneous diagonalization of the matrices in :

. Let be a positive integer. For any set of commuting normal matrices, there exists a unitary , such that for any , there exist for such thatTheorem

Nice post!

Thank you!