Chapter 5.5, Problem 23E

Chapter 7 will focus on matrices A with the property that A^T = A. Exercises 23 and 24 show that every eigenvalue of such a matrix is necessarily real.

23. Let A be an n × n real matrix with the property that A^T = A, let x be any vector in ℂⁿ, and let $q={\bar{\text{x}}}^{T}A\text{x}$. The equalities below show that q is a real number by verifying that $\bar{q}$ = q. Give a reason for each step.

$\begin{array}{l}\bar{q}=\stackrel{¯}{{\bar{\text{x}}}^{T}A\text{x}}={\text{x}}^T\overline{A\text{x}}={\text{x}}^{T}A\bar{\text{x}}={(}^{x^T}={\bar{\text{x}}}^T{A}^T\text{x = }q\\ \text{ (a) (b) (c) (d) (e)}\end{array}$