It is well known that: Let $A$ be a symmetric matrix, if $x^TAx>0, \forall x\in R^n$, then $A$ is positive definite and all this eigenvalues are positive.
Now my question is: if there exists at least a vector $x\in R^n$, $x^TAx>0$; and there exists at least a vector $y\in R^n$, $y^TAy<0$, does $A$ has at least one positive eigenvalue and at least one negative eigenvalue?