# On the Spectral Theorem (from Linear algebra)

The form we prefers says that every bounded self-adjoint operator is a multiplication operator. This means that given a bounded self-adjoint operator on a Hilbert space ${\mathcal{H}}$, we can always find a measure ${\mu}$ on a measure space ${M}$ and a unitary operator ${U: \mathcal{H} \rightarrow L^2(M, d\mu)}$ so that

$\displaystyle (UAU^{-1}f)(x) = F(x)f(x),$

for some bounded real-valued measurable function ${F}$ on ${M}$.

In the case of real, the Hilbert space $\mathcal{H}$ becomes a Euclidean space with an inner product. So an operator in $\mathcal{H}$ corresponds with an orthogonal matrix $A$. And an orthogonal matrix can be diagonalized, i.e. there exists $\lambda_1, \dots, \lambda_n \in \mathbb{R}$ and $S^{-1} = S^T$ such that

$\displaystyle S^TAS = diag(\lambda_1, \dots, \lambda_n)$.

(cont.)