Physics Problems & Solutions: Quantum Mechanics

The matrix $A= \left| \begin{array}{ccc} 0 & 1 & 0\\ 0 & 0 & 1 \\ 1 & 0 & 0 \end{array} \right|$ has 3 eigenvalues $\lambda_i$ defined by $Av_i=\lambda_i v_i$ . Which of the following statements is NOT true?

A. $\lambda_1 + \lambda_2 + \lambda_3 = 0$
B. $\lambda_1, \lambda_2,$ and $\lambda_3$ are all the real numbers
C. $\lambda_1\lambda_2 = + 1$ for some pair roots
D. $\lambda_1\lambda_2 + \lambda_2\lambda_3 + \lambda_3\lambda_1= 0$
E. $\lambda_i^3 =+1, i =1,2,3$

(GR9277 #98)

Solution:

Every real symmetric matrix is Hermitian, and therefore all its eigenvalues are real.

A symmetric matrix is a square matrix that is equal to its transpose.
$A=A^T \rightarrow a_{ij} = a_{ji}$

Matrix $A = \left| \begin{array}{ccc} 0 & 1 & 0\\ 0 & 0 & 1 \\ 1 & 0 & 0 \end{array} \right|$

is not symmetric, since $a_{ij} \neq a_{ji}$ . Therefore, its eigenvalues are NOT real.

Answer: B

Physics Problems & Solutions

Quantum Mechanics - Hermitian Matrix

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