Will a positive definite matrix always be diagonalizable, in such a way that we can only use ordinary eigenvectors?
Two other similar questions are:
$1.$ Will an $n \times n$ positive definite matrix always have $n$ distinct eigenvalues?
$2.$Can any eigenvalue of a positive definite matrix (with multiplicity $m \gt 1$) be "missing" an eigenvector?
Assuming that these are indeed valid questions.
No, consider, e.g., the matrix $$A = \begin{pmatrix}1 & 1 \\ 0 & 1 \end{pmatrix}.$$ Then $A$ is positive definite, but not diagonalizable.
The same example answers questions 1. and 2.