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!

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

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