[Spectral Theorem for Finite-Dimensional Hilbert Spaces] Let H be a finitedimensional Hilbert space over C with dim(H) = n < ∞, and let A : H → H
be a Hermitian operator, i.e., A = A† .
Then:
1. There exists an orthonormal basis {v1 , v2 , . . . , vn } of H consisting of eigenvectors of A.
2. The corresponding eigenvalues λ1 , λ2 , . . . , λn are real.
3. A admits a spectral decomposition
A=
m
X
λ k Pk ,
k=1
where λ1 , . . . , λm are the distinct eigenvalues of A, and Pk is the orthogonal projection onto the eigenspace
Ek = {v ∈ H | Av = λk v}.
These projections satisfy:
Pk Pj = δkj Pk ,
m
X
Pk = I,
Pk = Pk† .
k=1
Equivalently, in the orthonormal eigenvector basis, A is represented by a diagonal matrix with real entries.
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