Is symmetric matrix has nonzero eigen values or is Positive (semi-)definite?

Question

Let M be symmetric matrix. Its eigen values are 0 or real(but not complex)

$Mx = \lambda x$

taking transpose on both sides and multiplying by $Mx$ on both sides

$(Mx)^TMx = \lambda x^TMx, $

Let y = Mx then we have

$\lambda x^TMx = (Mx)^TMx = y^Ty$ --> this is nothing but squared sum. It means $y^Ty \geq 0$

It meaans $\lambda x^TMx \geq 0 \implies \lambda \geq 0 , $ OR $x^TMx \geq 0$ which means symmetric matrix has either eigen values non-negative or positive semi-definite? May i know is this true? I was thinking of some other problem but got this which i do not know

In defination of positive (semi-)definiteness, there are 2 conditions

1. symmetric matrix

2. $x^TMx \geq 0$

Answer 1

The reason that the eigen values of symmetric matrices are real is the following:

$\lambda=\langle v,\lambda v\rangle= \langle v,Av\rangle=\overline{\langle Av,v\rangle}=\langle \lambda v,v\rangle=\overline{\lambda}$