I am new here. I hope you guys can help me. Thanks a lot already.
First, let's do some setup. Let $n\in \mathbb{N}$ and let $v_{1},v_{2}, \ldots, v_{n}$ be an orthonormal basis of $\mathbb{R}^{n}$.
I am investigating a matrix of the Form, $M = \begin{bmatrix} m_{12}+m_{1n} & -m_{12} & 0 & \ldots & 0 & -m_{1n} \\ -m_{12} & m_{12}+m_{23} & -m_{23} & \ldots & 0 & 0 \\ \vdots & \ddots & \ddots & \ddots & \vdots & \vdots \\ 0 & 0 & \ldots & -m_{n-2,n-1} & m_{n-2,n-1}+m_{n-1,n} & -m_{n-1,n} \\ -m_{1n} & 0 & \ldots & 0 & -m_{n-1,n} & m_{1n}+m_{n-1,n} \end{bmatrix}$.
In particular, I am investigating the following system of equations induced by $M$ and the $v_{j}$'s. For $i \in \{ 1,2, \ldots, n\}$,
$$ v_{i}^{*}Mv_{i}=b_{i}.$$
I am curious for which vector $\mathbf{b}=(b_{1},b_{2}, \ldots, b_{n})$ this system of equations. I would imagine there is a solution for all $\mathbf{b}$ since I have $n$-degrees of freedom (choose of $m_{ij}$)on the left side and $n$ equations in total. But can we relate the $m_{ij}$'s to the $b_{i}$'s in a meaningful way? Or can I say something about the rank of this system?
Since, all rows of $M$ sum up to $0$, we know that $0$ must be an eigenvalue of $M$ with corresponding eigenvector $(1,1, \dots,1)$. Therefore $M$ does not have full rank. Does this mean that the system of equations also does not have full rank? This should not be an issue, since even if the system does not have full rank we have $n$ degrees for $n-1$ equations.
I also tried to convert the system into a matrix notation, $$ B\begin{bmatrix} m_{12} \\ m_{23} \\ \vdots \\ m_{1n} \end{bmatrix}=\begin{bmatrix} b_{1} \\ b_{2} \\ \vdots \\ b_{n} \end{bmatrix},$$ where $B \in \mathbb{R}^{n \times n}$, but I could not find a "nice" form for $B$.
I observed that $M$ is a Laplace matrix of a weighted undirected circular graph. I looked at some literature on Laplace matrices, but I could not find anything useful. Is there in general area in linear algebra that deals with systems of linear equations induced by a quadratic form? I mean systems of the form, $$ x_{i}^{*}Ax_{i}=a_{i}, $$ for $i \in \{1,2,\ldots, n \}$ where $x_{1},x_{2}, \ldots, x_{n}$ is a set of vectors and the entries of $A$ (the $A_{ij}$'s) are the variables.