I have the following problem. Given some vectors $\vec{w}_i\,, \vec{q} \in \mathbb{R}^3$ with $\|\vec{w}_i\|,\|\vec{q}\|\neq 0$ and a 3x3 real matrix $M$, find the solution to the set of equations $$ M\vec{d}_1-t_1\vec{q}+\kappa\vec{w}_1 = M \hat{e}_1 \,,\\ M\vec{d}_2-t_2\vec{q}+\kappa\vec{w}_2 = M \hat{e}_2 \,,\\ M\vec{d}_3-t_3\vec{q}+\kappa\vec{w}_3 = M \hat{e}_3 \,, $$ for some real numbers $\kappa, t_1, t_2, t_3$ and real vectors $\vec{d}_i\in \mathbb{R}^3$. Moreover, $\hat{e}_j$ denote the unit vectors in some orthonormal system.
Question 1: What are the conditions for solutions to exist?
Question 2: What is the general theory for such coupled matrix equations? How do I formally write down the solutions?
Attempt at solution/observations: Note that $\vec{d}_i=\hat{e}_i\,, \, \kappa=t_i=0 \, \forall i$, is a particular solution to the equations. Thus, one needs to find the solutions to the three homogeneous equations $$ M\vec{d}_i-t_i\vec{q}+\kappa\vec{w}_i = 0 \,, $$ for $i=1,2,3$. Then, the general solution would be the particular solution plus the solutions to the homogeneous equations.
Formally I can there write, $$ M \vec{d}_j = \kappa\vec{w}_j- t_j \vec{q}\equiv \vec{b}_j $$ and apply the theory of linear matrix equations.
In short, for an equation of the form $A\vec{x}=\vec{b}$, solutions exist if $$ (A A^\dagger-\mathbb{I})\vec{b} =0 \,, $$ where $A^\dagger$ denotes the pseudoinverse of the matrix $A$. If the last equations holds, then the solutions are given by $$ \vec{x} = A^\dagger \vec{b} - (\mathbb{I}-A^\dagger A) \vec{c} \,, $$ where $\vec{c}$ is any real vector.
So, if I am allowed to do this trick, I can formally write the general solutions to my equations as $$ \vec{d}_j = \hat{e}_j + M^\dagger \vec{b}_j - (\mathbb{I}-M^\dagger M) \vec{c} \,, \\ \qquad \qquad = \hat{e}_j + \kappa M^\dagger\vec{w}_j- t_j M^\dagger \vec{q} - (\mathbb{I}-M^\dagger M) \vec{c} \,, $$ as long as $$ (M M^\dagger-\mathbb{I})(\kappa\vec{w}_j- t_j \vec{q}) =0 \,. $$ However, I am not sure if my approach is correct.
Question 3: Is my formulation and approach to the problem correct? Particularly, the equations seem to have the almost the form $A\vec{x}=\vec{b}$ but not quite. Any advice how to proceed?
Thank you in advance!
You have illustrated a feasible solution, hence the system is always consistent.
Your equations can be written as the following aug
$$\begin{bmatrix} M & 0_{3 \times 3} & 0_{3 \times 3} & -q_1 & 0_{3 \times 1} & 0_{3 \times 1} & w_1 \\ 0_{3 \times 3} & M & 0_{3 \times 3} & 0_{3 \times 1} & -q_2 & 0_{3 \times 1} & w_2 \\ 0_{3 \times 3} & 0_{3 \times 3} &M & 0_{3 \times 1} & 0_{3 \times 1} & -q_3& w_2 \end{bmatrix}\begin{bmatrix} d_1 \\ d_2 \\ d_3 \\ t_1 \\ t_2 \\ t_3 \\ \kappa\end{bmatrix}=\begin{bmatrix} Me_1 \\ Me_2 \\ Me_3\end{bmatrix}$$
Now, this reduces to a classical linear system of equations.