Which one of the following statements is NOT true for a square matrix $A$?
- If $A$ is upper triangular, the eigenvalues of $A$ are the diagonal elements of it
- If $A$ is real symmetric, the eigenvalues of $A$ are always real and positive
- If $A$ is real, the eigenvalues of $A$ and $A ^T\:\text{(A transpose)}$ are always the same
- If all the principal minor of $A$ are positive, all the eigenvalues of $A$ are also positive