Convex Relaxation Problem (theory )

Question

I was wondering if any of you folks could help me with this optimization problem. I want to show if I find a solution $X^*$ for $(P'')$ for which $rank(X ^* ) = 2$, then $X ^* = uu^T $. Here is the problem $(P'')$ $$ \underset{u\in\mathbb{R}^{N+2 , 2} , X \in \mathbb{S}^{N+2}}{\min} \Sigma u[:,1]\\ X_{i,i} - 2X_{i,i+1} +X_{i+1 ,i+1}- L^2 = 0 \quad \text{for} \quad i \in [0, ... , N]\\ u[0, :] = (0, 0) ,\ u[N+1 , :] = (a,b) ,\ X_{0,0} = 0 ,\ X_{N+1 ,N+1} = a^2 + b^2 \\ \begin{pmatrix} I_2 & u^T \\ u & X \end{pmatrix}\ge 0 $$ Thank you for your help.