# Making a matrix positive definite

Suppose $$p \times p$$ matrix $$X$$ is symmetric and not positive definite (PD). I want to find the minimum value for $$\mu$$ such that $$X + \mu I_p$$ is positive definite. To this end, I suppose we can solve a semidefinite program (SDP) of the form

$$\begin{array}{ll} \text{minimize} & \mu\\ \text{subject to} & X + \mu I_p \succeq 0\\ & \mu \geq 0\end{array}$$

where $$A \succeq 0$$ denotes that matrix $$A$$ is PD, and $$I_p$$ is the $$p \times p$$ identity matrix.

Is there a more elegant approach to find $$\mu$$ using linear algebra? I am also curious if $$\mu$$ has any interpretation or not.

Thanks for possible feedback!

• Just let $\mu$ to be the absolute value of the smallest eigenvalue of $X$ (since $X$ is not PD, this must be $\geq 0$). – Wanshan Jun 1 at 20:14
• @Wanshan, thanks. this makes sense. – user2512443 Jun 1 at 20:27
• Are you assuming $X$ is symmetric? – kimchi lover Jun 1 at 20:32
• @kimchilover Yes. Thanks for this important note. Edited the question. Does this change anything? – user2512443 Jun 1 at 20:34
• If the matrix is symmetrical then you have real eigenvalues which makes it possible to apply Wanshan's idea. – OneAndOnlyDaniel Jun 1 at 20:50