restrictions in order a matrix be positive definite

Question

I have a 2x2 matrix and i want to find the restrictions in order to be positive definite. I wrote that matrix should be symmetrical and the determinants of the leading principal minor matrices should be all of them positive (>0). Is there anything else i should mention??

Answer 1

We can equivalently say that for a matrix to be positive definite, the eigenvalues of a must be greater than zero.

Now Vieta's formulas give us the following characteristic equation for a general $2 \times 2$ matrix $A$

$$ p(\lambda) = {\lambda}^2 - \text{tr}(A)\lambda + \det{A} $$

Using the quadratic formula we have

$$ \lambda = \frac{\text{tr} \, A \ \pm \sqrt{(\text{tr} \, A)^2 - 4\det{A}}}{2} $$

From this formula can you tell what conditions must hold for both roots to be positive?