Does $x^T(M+M^T)x \geq 0 \implies x^TMx \geq 0$ hold in only one direction? I know this is true for the "if" part, but what about the "only if"?
Can you give me one example when the "only if" part does not hold? I am not quite sure about this.
I forgot to tell you that $M$ is real and $x$ arbitrary.

Well
$x^{T}Mx \geq 0 \implies x^{T}M^{T}x \geq 0$ (by taking transpose)
Hence, $x^{T}(M^{T} + M)x \geq 0$.
So, this gives you the one side. However, you wrote the converse in your title.
