is the sum of positives definite matrix,still positive definite?

Question

please give me hints on A = B + C. How can we show that A is positive definite?

Answer 1

The associated hermitian form is $$x^HAx=x^HBx+\sum_{i=1}^n c_{ii}\overline{x}_ix_i\ge x^HBx>0$$ as soon as $x\ne 0$.