Given the following under-determined linear system:
$$ \begin{bmatrix} 1 & 0 & 1 \\ 1& 1 & 0.5 \end{bmatrix} \begin{Bmatrix} \ v \\ x \\ f \end{Bmatrix} = \begin{Bmatrix} \ 0 \\ 1 \end{Bmatrix} $$
Write the general solution of the above system as a combination of a particular solution and the null vector of A: $$ \mathbf{z} = \mathbf{z}_p + \alpha\mathbf{w}$$ where, $\mathbf{w} \in \mathscr{N}(A)$
This part is straightforward and can be easily found to be a solution of the form: $$ \mathbf{z} =\begin{Bmatrix} \ 1 \\ 1/2 \\ -1 \end{Bmatrix} + \alpha \begin{Bmatrix} \ 2/3 \\ -1/3 \\ -2/3 \end{Bmatrix} $$ Now, I wish to find the optimal $\alpha$ that identifies the minimum norm solution of the original equation and compare it to the solution obtained via: $$\mathbf{z}^* = \mathbf{A}^{\text{T}}(\mathbf{AA}^{\text{T}})^{-1}\mathbf{y}$$
I know the condition for optimality is when the gradient of the cost function (norm) is zero. i.e:$$\frac {\partial J}{\partial \alpha} = 0$$Or(?, not completely sure about this equation), $$\alpha^* = \arg\min (J(\mathbf{x})) = \mathbf{x}^{\text{T}}\mathbf{x} = (\mathbf{x}_p + \alpha \text{n})^{\text{T}}(\mathbf{x}_p + \alpha \text{n})$$
My question is:
How do you take the partial derivative of the above matrix equation? Is it as simple as writing the system of equations and then taking the partial and setting equal to 0? Should I expect to be able to solve for $\alpha$ easily since everything else is known?
Thanks!