Trace of a Summation of Positive Definite Matrices with a Real Coefficient

Question

Show that $$\mbox{tr} \left( \sum_{i=1}^k (\alpha A_{i} + B_{i})^2 \right) \geqslant 0, \qquad \forall \alpha \in \mathbb{R}$$ given that $A_{i}$ and $B_{i}$ are positive definite $n \times n$ complex matrices $\forall i=1,\dots,k$.

Any hint?

Answer 1

Hint: $(\alpha A_i + B_i)^2$ is positive semidefinite.