Eigenvectors of symmetric matrices are real???

Question

I am studying about symmetric matrices. I find out a lot of books state the theorem of existence of real eigenvector of symmetric matrices but didn't provide any solution. I am wander this problem.

Prove that all the eigenvectors of symmetric matrices are real.

Answer 1

It is not necessarily true that eigenvectors of real symmetric matrices are real; what is true, however, is that every real eigevalue has a real eigenvector, and since every eigenvalue of a symmetric matrix is real, a complete set real eigenvectors may be found. For if

$Ax = \lambda x, \tag 1$

then taking complex conjugates yields

$A \bar x = \lambda \bar x; \tag 2$

adding (1) and (2),

$A(x + \bar x) = \lambda(x + \bar x); \tag 3$

if

$x + \bar x = 0, \tag 4$

then

$\bar x = -x, \tag 5$

that is, $x$ is purely imaginary, hence $ix$ is real, and from (1),

$A(ix)= \lambda(ix), \tag 6$

so $\lambda$ has a real eigenvector; on the other hand, if

$x + \bar x \ne 0, \tag 7$

then $x + \bar x$ is a real eigenvector associated to $\lambda$.