If eigenvalues are positive, is the matrix positive definite? If the matrix is positive definite, then all its eigenvalues are strictly positive.  

Is the converse also true?
That is, if the eigenvalues are strictly positive, then matrix is positive definite?
Can you give example of $2 \times 2$ matrix with $2$ positive eigenvalues but is not positive definite?
I think this is false. Let $A = \begin{pmatrix} 1 &amp; -3 \\ 0 &amp; 1 \end{pmatrix}$ be a 2x2 matrix, in the canonical basis of $\mathbb R^2$. Then A has a double eigenvalue b=1. If $v=\begin{pmatrix}1\\1\end{pmatrix}$, then $\langle v, Av \rangle &lt; 0$.
The point is that the matrix can have all its eigenvalues strictly positive, but it does not follow that it is positive definite.
For a matrix to be positive definite:

1) it must be symmetric
2) all eigenvalues must be positive
3) it must be non singular
4) all determinants (from the top left down the diagonal to the bottom right - not jut the one determinant for the whole matrix) must be positive.

If a 2x2 positive definite matrix is plotted it should look like a bowl. If the matrix is singular then it's a trough which follows the vector which takes the matrix to zero. If the sum of the cross terms is bigger than the trace then the plot is a saddle at zero.