# A question on PSD (positive semidefinite) completion

Let $$A=\left[\begin{array}{ccccccc} |a_0|^2 &a_0\bar{a}_1 &a_0\bar{a}_2 &0 &a_0\bar{a}_3 &0&0\\ \bar{a}_0a_1 &|a_1|^2 & c &0 &a_1\bar{a}_3 &0&0\\ \bar{a}_0a_2 &\bar{c} &|a_2|^2 &0 &a_2\bar{a}_3 &0&0\\ 0 &0 &0 &0 &0 &0&0\\ \bar{a}_0a_3&\bar{a}_1a_3 &\bar{a}_2a_3 &0 &|a_3|^2 &0&0\\ 0 &0 &0 &0 &0 &0 &0 \\ 0 &0 &0 &0 &0 &0 &0 \end{array}\right]$$.

We note that if we choose $$c=a_1\bar{a}_2$$, then $$A=b^\dag b,\mbox{ with } b=(a_0~ a_1~ a_2~ 0~ a_3~ 0 ~0)^T$$ and hence A is PSD. Now my guess is there does not exists any other choice for c to make A PSD i.e., A is PSD iff $$c=a_1\bar{a}_2$$. Can anyone help me to prove (or give a counter example to ) this?

One can assume the additional condition Tr(A)=1. To give a counterexample, one can assume as many of $$a_i,\quad i=0,1,2,3$$ to be zero. Please help me.