# Thread: Spectral Theorem

1. ## Spectral Theorem

I am trying to understand these theorems from linear algebra. I never learned this sort of stuff but I think I understand the theorems. Can you tell me if my understanding is correct?

1)Let $\displaystyle A$ be a $\displaystyle n\times n$ matrix over $\displaystyle \mathbb{R}$. If $\displaystyle A$ is symmetric then $\displaystyle A$ can be diagnolized i.e. written in the form $\displaystyle A = MDM^{-1}$ where $\displaystyle D$ is a diagnol matrix. But it turns out that not only $\displaystyle A$ is diagnolizable it is orthogonally diagnolizable i.e. $\displaystyle A = NDN^{-1}$ where $\displaystyle N$ is an orthogonal matrix (defined to be such that $\displaystyle N^{-1} = N^T$). This result is a major result known as the "Spectral Theorem". But then as a consequence of this diagnolization we see that its eigenvectors form an orthogonal basis.

2)How about making #1 more abstract? Let $\displaystyle V$ be a finite dimensional inner product space over a field $\displaystyle F$. Let $\displaystyle T:V\to V$ be a linear transformation. We say that $\displaystyle T$ is a "symmetric transformation" when its matrix relative to the basis is a symmetric matrix. Then if $\displaystyle T$ symmetric its eigenvectors form an orthogonal basis for $\displaystyle V$.

2. (2) is fine, but since V is a finite dimensional inner product space, it makes no difference if it is only F^{dimension}.

Spectral theorems are far too important for noone to try to generalize There is a version about self-adjoint operators in infinite dimensional Hilbert spaces (which is a direct generalization of the finite-d case, and is used very finely in proving existence of solutions for the Laplacian operator) and a very unexpected version about "completely continuous" operators in the same setting, which leads to the so-called Functional Calculus.

Alright, the thing is Spectral Theorems are still being worked on!

3. The notion of symmetry seems to be special to the real scalars. If you want a spectral theorem for complex-valued matrices, for example, then the relevant property is not symmetry but selfadjointness. (A complex matrix is selfadjoint is it is equal to its conjugate transpose.) A selfadjoint complex matrix is not merely diagonalisable but unitarily diagonalisable. In other words, it is diagonalisable by a unitary matrix (namely a matrix whose inverse is its conjugate transpose). That is the complex version of the spectral theorem.

For other fields, I doubt whether there is any useful analogue to the concept of symmetry. For example, the 3 by 3 matrix over the field of 3 elements given by

0 1 1
1 2 1
1 1 0

is symmetric but not diagonalisable.

[Sorry about the primitive formatting. The forum's server is still having troubles, and the LaTeX facility isn't working.]

4. Originally Posted by Opalg
For other fields, I doubt whether there is any useful analogue to the concept of symmetry.
In my field theory book there is an exercise and it says to assume the fact "that every symettric matrix can be diagnolized by an orthogonal transformation". I assume it means A = MDM* where M* is the transpose and also the inverse. The problem is stated in general terms: over any field with vector space F^n. If there is no generalization, then why does the book assume that?