Principles of Quantum Mechanics · Problem 1.8.6

Mathematical Introduction — R. Shankar

Physics — Quantum Mechanics · eigenvalues-determinant-trace

Problem

(1) We have seen that the determinant of a matrix is unchanged under a unitary change of basis. Argue now that det⁡Ω=product of eigenvalues of Ω=∏i=1nωi\det \Omega = \text{product of eigenvalues of } \Omega = \prod_{i=1}^{n} \omega_i for a Hermitian or unitary Ω\Omega. (2) Using the invariance of the trace under the same transformation, show that Tr Ω=∑i=1nωi\text{Tr}\, \Omega = \sum_{i=1}^{n} \omega_i

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