Introduction to Quantum Mechanics · Problem A.26

Afterword — David J. Griffiths

Physics — Quantum Mechanics · eigenvectors-eigenvalues

Problem

Consider the matrices

\mathsf{A} = \begin{pmatrix} 2 & 2 & -1 \\ 2 & -1 & 2 \\ -1 & 2 & 2 \end{pmatrix}, \quad \mathsf{B} = \begin{pmatrix} 2 & -1 & 2 \\ -1 & 5 & -1 \\ 2 & -1 & 2 \end{pmatrix}.

(a) Verify that they are diagonalizable and that they commute. (b) Find the eigenvalues and eigenvectors of

\mathsf{A}

and verify that its spectrum is degenerate. (c) Are the eigenvectors that you found in part (b) also eigenvectors of

\mathsf{B}

? If not, find the vectors that are simultaneous eigenvectors of both matrices.