## 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.

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