Is a matrix that is symmetric and has all positive eigenvalues always positive definite? I know a symmetric matrix is positive definite if and only if every eigenvalue is positive. However, is a matrix that is symmetric and has all positive eigenvalues always positive definite? More specifically, I have the following matrix:
$$\begin{bmatrix}3& -1 \\-1 & 3 \end{bmatrix}$$
Its eigenvalues are $4$ and $2$, and it is symmetric. Is it positive definite? Thanks.

Yes in the real case. 
Let $ A $ be a matrix that is symmetric and has all positive eigenvalues. Then,  $\forall x\in V, x^TAx > 0 $. This is true for  all eigenvectors $ x_i $ of $ A $. Hence $0<x^T_iAx_i=x_i\alpha_i x_i=||x_i||^2\alpha_i $, so $\alpha_i> 0 $.
Suppose $\alpha_i> 0\forall i $.  Then, on eigenvectors, $0<x^T_iAx_i$. $ A $ is real and symmetric.  Hence, in $ V $ there exists an orthonormal basis of $ A $, by the spectral theorem. In this basis, $\forall y\in V, y=\sum a_ix_i $. The above relation thus gives $ y^TAy=\sum a_i^2\alpha_i> 0 $. Q.E.D.
