Symmetric rank two correction of a symmetric positive definite matrix

Question

Let $P$ is a symmetric positive definite matrix and define $P'$ as follow:

$$P'=P + (e_ie_j^T+e_je_i^t)P_{ij}$$

(where $i \neq j$).

(so $P'$ is a rank 2, symmetric correction for $P$)

Is $P'$ is positive definite as well?

Any comment or reference is highly appreciated.

Answer 1

Counterexample. $$ P \; = \; \left( \begin{array}{rr} 5 & -4 \\ -4 & 5 \end{array} \right) $$ $$ P' \; = \; \left( \begin{array}{rr} 5 & -8 \\ -8 & 5 \end{array} \right) $$

Another counterexample. $$ P \; = \; \left( \begin{array}{rr} 7 & 6 \\ 6 & 7 \end{array} \right) $$ $$ P' \; = \; \left( \begin{array}{rr} 7 & 12 \\ 12 & 7 \end{array} \right) $$