$\mathbf{x_1},\mathbf{x_2}$ are known and I need to solve the following objective wrt to one variable $\mathbf{y}$. The single constraint is $y(1,1)=1$. This is expressed as an inner product $\mathbf{y^T}\mathbf{e_y}=1$
\begin{align} \mathbf{e_y} = \left[ \begin{array}{ccc} 1 \space\space 0 \space\space 0 \space\space \cdots 0 \end{array} \right]^T \end{align}
\begin{equation} \begin{array}{c} \underset{\mathbf{y}}{\text{minimize}} \hspace{4mm} \big(\left( \mathbf{y}^T \mathbf{x_1} \right)^2 + \left( \mathbf{y}^T \mathbf{x_2} \right)^2\big) \\ s.t \hspace{10mm} \hspace{17mm}\mathbf{y}^T\mathbf{e_y} = 1, \end{array} \end{equation}
Rewriting cost function
\begin{align} J &= \left( \mathbf{y}^T \mathbf{x_1} \mathbf{x_1}^T \mathbf{y} \right) + \left( \mathbf{y}^T \mathbf{x_2} \mathbf{x_2}^T \mathbf{y} \right) \\ &= \mathbf{y}^T \left( \mathbf{x_1} \mathbf{x_1}^T + \mathbf{x_2} \mathbf{x_2}^T \right)\mathbf{y} \\ &= \mathbf{y}^T \mathbf{Z}\mathbf{y} \end{align}
$J=0$ if and only if $\mathbf{y}_{opt}\in null(\mathbf{Z})$. But if $\mathbf{Z}$ is full rank then $J\neq 0$. Is this interpretation correct?
I cannot figure out how to search for solution when null space solution is not possible. Can someone please help me with this. I have a single constraint and it is crucial in my original problem