The definition of positive definite function is something like:
for all nonzero $x\in R^n$ then the quadratic function $V(x)=x^TPx>0$ where $P \in R^n$x$R^n$ is a positive-definite matrix.
for example when I am studing the vector $x=[x_1,x_2]^T$ , the quadratice chosen function $V=0.5(x_1^2+x_2^2)>0$ is a positive definite function.
But in one textbook I have found that for a system with three state, i.e $x\in R^3$ for example $x=[x_1,x_2,x_3]^T$ if we choose the quadratic function as :
$V=0.5(x_1^2+x_2^2)$ then this chosen function is positive semi-definite in $R^3$.
I can't understand why the last function is positive semi-definite in $R^3$ and not positive definite? I thought that because it has been taken only two states i.e $x_1,x_2$ and it has been dropped $x_3$. But I didn't understand the idea of that.