Principles of Quantum Mechanics · Problem 1.8.4

Mathematical Introduction — R. Shankar

Physics — Quantum Mechanics · eigenvalue-problem-non-hermitian

Problem

An arbitrary

n \times n

matrix need not have

n

eigenvectors. Consider as an example

\Omega = \begin{bmatrix} 4 & 1 \\ -1 & 2 \end{bmatrix}

(1) Show that

\omega_1 = \omega_2 = 3

. (2) By feeding in this value show we get only one eigenvector of the form

\frac{1}{(2a^2)^{1/2}}\begin{bmatrix} +a \\ -a \end{bmatrix}

We cannot find another one that is LI.