Spectrum of a Self-Adjoint Operator in Finite Dimension

Let $H$ be a finite-dimensional complex Hilbert space (Complex Hilbert Space Finite) and let $A:H\to H$ be self-adjoint (Self Adjoint Operator Observable).

The spectrum of $A$ is the set of scalars $\lambda\in\mathbb C$ for which $A-\lambda I$ is not invertible. In finite dimension this is exactly the set of eigenvalues of $A$ , and self-adjointness makes every spectral value real.

Spectral theorem (finite-dimensional form)

There exist distinct real eigenvalues $\lambda_1,\dots,\lambda_m$ and orthogonal projections $P_1,\dots,P_m$ onto the corresponding eigenspaces such that:

$P_iP_j = 0$ for $i\ne j$ ,
$\sum_{i=1}^m P_i = I$ ,
$A$ decomposes as $A = \sum_{i=1}^m \lambda_i P_i.$

The projections $P_i$ are uniquely determined by $A$ (they are the spectral projectors).

Functional calculus

For any function $f$ defined on the spectrum $\{\lambda_i\}$ , one defines

f(A) := \sum_{i=1}^m f(\lambda_i)\,P_i.

Common examples include powers $A^k$ , the exponential $e^{A}$ , and $\log(A)$ when $A$ is positive definite.

Quantum interpretation

If $A$ is an observable, then the possible measurement outcomes are its spectral values $\lambda_i$ . In a state $\rho$ (Density Operator), the Born rule assigns outcome probabilities

\Pr(\lambda_i) = \operatorname{Tr}(\rho P_i),

using the operator trace (Trace Operator).