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## Lecture (10 minutes - 40 minutes with proofs)

- Hermitian matrices have real eigenvalues.
- Hermitian matrices have one eigenvector for each eigenvalue (except for degeneracy).
- The eigenvectors of a Hermitian matrix are orthogonal and can be normalized,
*i.e.*they are orthonormal. - The eigenvectors of a Hermitian matrix form an orthonormal basis for the space of all vectors in the vector space.
- Commuting operators share the same eigenbasis.

Notes for this lecture:

- Use bra-ket notation for proofs.
- Refer to specific examples from eigenvectors/eigenvalues activity.
- (Optional) Refer to example of Fourier series.