# least-squares estimate for simple 3x3 singular matrix with constraints

I have some measurements $M_1, M_2, M_3$ and some parameters $a,b,c$ I want to estimate. They are related as follows:

$$\begin{bmatrix}M_1 \\ M_2 \\ M_3\end{bmatrix} = \begin{bmatrix}1 & -1 & 0 \\ 0 & 1 & -1 \\ -1 & 0 & 1\end{bmatrix} \begin{bmatrix}a \\ b \\ c\end{bmatrix}$$

where the 3x3 matrix is singular (no information about $a+b+c$), but I also know that $a \approx 1$ and $b \approx 0$ and $c \approx 0$, so it seems like I would want to minimize $(a-1)^2 + b^2 + c^2$.

How would I solve for $a,b,c$ in a least-squares sense, given this information? I know how to use least-squares estimates without constraints, in an overdetermined system, but not sure what to do here.

$$\arg \min \left\| A x - b \right\|_{2}^{2} + \lambda \left\| x - d \right\|_{2}^{2}$$
Where $A$ is the matrix above, $b$ is the measurements vector and $d = \left[ 1, 0, 0 \right]^{T}$.
The parameter $\lambda$ states you assurance in the model you supplied, namely how close $x$ is to the vector $d$.
$$\hat{x} = \left( {A}^{T} A + \lambda I \right)^{-1} \left( {A}^{T} b + \lambda d \right)$$
As can be seen for $\lambda = 0$ the solution will be the least squares solution and for $\lambda \to \infty$ you'll get $\hat{x} = d$.