Lecture 57. Spectral Theorem скачать в хорошем качестве

Lecture 57. Spectral Theorem

In this lecture we prove the Spectral Theorem for self-adjoint operators. In real case this becomes a spectral theorem for symmetric matrices. We prove that a complex self-adjoint (or real symmetric) operator has an orthonormal basis consisting of eigenvectors with real eigenvalues. We show how to use this theorem to diagonalize quadratic forms. Finally, we explain why this theorem is important for quantum mechanics. This is a lecture in a course on "Linear Algebra" for students specializing in mathematics. 0:00 Maximum and minimum values of quadratic forms 7:28 Orthogonal and unitary matrices 10:10 Self-adjoint operators 17:24 Eigenvalues of self-adjoint operators are real 20:22 Eigenvectors of self-adjoint operators are orthogonal to each other 23:17 Statement of the Spectral Theorem 27:20 Proof of the Spectral Theorem 38:14 Example: using Spectral Theorem to diagonalize a quadratic form 52:16 Axiomatics of Quantum Mechanics and the Spectral Theorem