If A is positive semidefinite, then $A+\alpha I$ with $\alpha\neq 0$ is positive definite?

Question

If $A$ is a symmetric positive semidefinite matrix, then $A+\alpha I$ with $\alpha> 0$ is positive definite?

Or are there some conditions to $\alpha$ so that it verifies?

$I$ is the identity matrix

Answer 1

$$x^T(A+\alpha I)x=x^TAx+\alpha x^Tx=x^TAx+\alpha||x||^2\geq\alpha||x||^2$$

Answer 2

You can also easily see this by diagonalising A.