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 ).

Spectrum in finite dimension

In finite dimension, the spectrum of $A$ is the set of scalars $\lambda\in\mathbb{C}$ for which $A-\lambda I$ is not invertible. Equivalently, the spectrum is exactly the set of Eigenvalue s of $A$.

For self-adjoint $A$, all eigenvalues (hence all spectral values) are 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:

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

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

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

using the operator trace (Trace Operator ).