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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?

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  • $\begingroup$ yes and it is fairly easy to see why: $\langle (A+D)x,x\rangle = \langle Ax,x\rangle + \langle Dx, x\rangle$ by the linearity of inner products. If $A$ is positive definite and $D$ is a diagonal matrix with nonnegative elements, both terms are larger than zero, i.e positive definite $\endgroup$
    – lmaosome
    Aug 18, 2021 at 21:30
  • $\begingroup$ @Imaosome the term $x^TDx$ can be zero for the special case $D=0$. $\endgroup$
    – user
    Aug 18, 2021 at 21:31

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We have that $\forall x\neq 0$

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

therefore your claim is true.

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