$\newcommand{\ones}{\mathbf 1}$

If $A\in {\mathbf R}^{n \times n}$ has all real eigenvalues and an orthonormal set of eigenvectors, then it is symmetric.

If $A\in {\mathbf R}^{n \times n}$ is symmetric, then all its eigenvalues are real and it has an orthonormal set of eigenvectors.

We can have $x^TAx = x^TBx$ for all $x \in {\mathbf R}^n$ with $A \neq B$.

If $A \in {\mathbf R}^{n \times n}$ is positive semidefinite, then $A_{ij} \geq 0$ for $i,j=1, \ldots, n$.

For any two matrices $A,B \in \mathbf{R}^{n \times n}$, either $A \geq B$ or $B \geq A$.