关于加权无向循环图拉普拉斯矩阵诱导的内积型线性方程组的解与参数关系的技术问询

Hi everyone, I'm new here and really hope you can lend me a hand—thanks so much in advance!

First, let's lay out the setup:

• Let $n\in \mathbb{N}$
• Let $v_{1},v_{2}, \ldots, v_{n}$ be an orthonormal basis of $\mathbb{R}^{n}$

I'm investigating a matrix of the following form:
$$
M = \begin{bmatrix}
m_{12}+m_{1n} & -m_{12} & 0 & \ldots & 0 & -m_{1n} \
-m_{12} & m_{12}+m_{23} & -m_{23} & \ldots & 0 & 0 \
\vdots & \ddots & \ddots & \ddots & \vdots & \vdots \
0 & 0 & \ldots & -m_{n-2,n-1} & m_{n-2,n-1}+m_{n-1,n} & -m_{n-1,n} \
-m_{1n} & 0 & \ldots & 0 & -m_{n-1,n} & m_{1n}+m_{n-1,n}
\end{bmatrix}
$$

Specifically, I'm looking at the system of equations induced by $M$ and the $v_j$'s. For each $i \in { 1,2, \ldots, n}$, we have:
$$ v_{i}^{*}Mv_{i}=b_{i}.$$

My Questions #

I'm curious about which vectors $\mathbf{b}=(b_{1},b_{2}, \ldots, b_{n})$ make this system solvable. My initial intuition is that there should be a solution for any $\mathbf{b}$, since we have $n$ degrees of freedom (the choices of $m_{ij}$) and exactly $n$ equations. But I'm wondering:

• Can we establish a meaningful, explicit relationship between the $m_{ij}$'s and the $b_{i}$'s?
• What can we say about the rank of this system?

Observations So Far #

I noticed that all rows of $M$ sum to 0, which means 0 is an eigenvalue of $M$ with corresponding eigenvector $(1,1, \dots,1)$. This tells us $M$ doesn't have full rank—but does this imply the system of equations also lacks full rank? I don't think this should be a showstopper, since even if the system has rank $n-1$, we still have $n$ degrees of freedom to satisfy those $n-1$ equations.

I also tried rewriting the system in matrix notation as:
$$ B\begin{bmatrix} m_{12} \ m_{23} \ \vdots \ m_{1n} \end{bmatrix}=\begin{bmatrix} b_{1} \ b_{2} \ \vdots \ b_{n} \end{bmatrix},$$
where $B \in \mathbb{R}^{n \times n}$, but I couldn't find a clean or structured form for $B$.

Background & Related Inquiry #

I realized $M$ is the Laplacian matrix of a weighted undirected circular graph. I've dug into some literature on Laplacian matrices but haven't found anything directly applicable to this problem.

Is there a general subfield of linear algebra that deals with systems of linear equations induced by quadratic forms? I'm thinking specifically about systems like:
$$ x_{i}^{*}Ax_{i}=a_{i}, $$
for $i \in {1,2,\ldots, n }$, where $x_{1},x_{2}, \ldots, x_{n}$ are a set of vectors and the entries of $A$ are the variables we're solving for.

备注：内容来源于stack exchange，提问作者SP1997