So I have a quadratic function in the form: $f(x) = x^T A x + bx $ where A is a positive definite matrix and symmetric $ n\times n$ matrix and $q\in\mathbb{R}^n$.
So I know know that the function is strict convex and has a global minimum. Furthermore, all eigenvalues of the matrix are larger than zero. I can visualize how the matrix is bounded from below, but how can I proof this?
I tried to prove it using:
$$f(tx + (1-t)y) = tf(x) + (1-t)f(y)$$
But do not know how to incorporate the general form into this in order to prove it.