2
$\begingroup$

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

$\endgroup$
0

2 Answers 2

2
$\begingroup$

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

$\endgroup$
0
0
$\begingroup$

You can also easily see this by diagonalising A.

$\endgroup$

Not the answer you're looking for? Browse other questions tagged .