Does adding non-negative values to diagonal of positive definite matrix preserves positive definiteness

Question

Does adding non-negative values to the diagonal of a positive definite matrix preserves its positive definiteness?

For example, $A$ is a symmetric positive definite matrix, and $D$ is a diagonal matrix with non-negative elements. Is $A+D$ positive definite?

Answer 1

We have that $\forall x\neq 0$

$$x^T(A+D)x = x^TAx+x^TDx > 0$$

therefore your claim is true.