Q:

Is there any difference between a non-singular, symmetric matrix and a positive definite matrix?

In optimization related context, you often see people impose the assumption that certain matrix is "symmetric and non-singular".
I wonder if this is the equivalent condition as the matrix being "positive definite".
It seems to be obvious, a positive definite matrix is symmetric (by standard assumption) and since all eigenvalues are real and positive, therefore non-singular.
At the same time, I know many characterizations of positive definite matrix, for instance, something about the determinants of the minors, eigenvalue characterization etc, but seldom have people mentioned that a matrix is positive definite matrix iff it is symmetric and non-singular. Maybe it is too obvious so people don't mention it. 
So is there any meaningful distinction between these two sets of matrices?

A:

The matrix $\begin{bmatrix}1&0\\0&-1\end{bmatrix}$ is symmetric and nonsingular but it is not positive definite because it has the eigenvalue $-1$.

