Chapter 5.5, Problem 24E

Let A be an n × n real matrix with the property that A^T = A. Show that if Ax = λx for some nonzero vector x in ℂⁿ, then, in fact. λ is real and die real part of x is an eigenvector of A. [Hint: Compute ${\bar{\text{x}}}^{T}A\text{x}$, and use Exercise 23. Also, examine the real and imaginary parts of Ax.]