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.)

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