TL;DR
Is there any way to solve the following matrix equation
$$ (L - \Lambda)Y = 0 $$
where $L \in \mathbb{R}^{n \times n}$, $Y \in \mathbb{R}^{n \times k}$ (with $k < n$) is a matrix of unknowns, $0$ is a zero matrix and crucially $\Lambda \in \mathbb{R}^{n \times n}$ is a diagonal matrix of $n$ Lagrange multipliers $\lambda_i$?
EDIT. We can also assume that $L$ is symmetric.
I want to find a Laplacian embedding of a graph constrained so all nodes are positioned on a unit sphere in $\mathbb{R}^k$. For simplicity, I assume that the graph is undirected, unweighted and has no self-loops.
Let $D$ be a diagonal matrix with node degrees along the diagonal, and $A = \{a_{ij}\}$ an adjacency matrix of a graph. Then we define the Laplacian matrix in the standard way as a symmetric matrix:
$$ L = D - A $$
Now, I assume that all nodes are embedded in $\mathbb{R}^k$, so there is a matrix $Y \in \mathbb{R}^{n \times k}$ of which $i$-th row gives the embedding coordinates of the $i$-th node of the graph.
So I try to find $Y$ by minimizing (squared) Euclidean distances between positions of connected nodes
$$ \sum_{i, j} a_{ij}||y_i - y_j||^2 $$ where $y_i$ is $i$-th row of $Y$ and subject to a constraint
$$ \forall_{i = 1, \ldots, n} \quad y_i^\top y_i = ||y_i||^2 = 1 $$ that is, I assume that all nodes are positioned on a unit sphere in $\mathbb{R}^k$.
This is equivalent to the following constrained optimization problem using matrix notation
$$ \text{argmin}\text{Tr}(Y^\top L Y) \quad \text{s.t.} \quad y_i^\top y_i = 1 \quad (i = 1, \ldots, n) $$
And this (I think) yields the following Lagranganian
$$ \begin{split} \mathcal{L} &= \text{Tr}(Y^\top L Y) - \sum_{i}\lambda_i(y_i^\top y_i - 1) \\ &= \text{Tr}(Y^\top L Y) - \text{Tr}(\Lambda YY^\top) + \text{Tr}(\Lambda) \end{split} $$ where $\Lambda \in \mathbb{R}^{n \times n}$ is a diagonal matrix of individual Lagrange multipliers $\lambda_i$.
After differentiating with respect to $Y$ we get
$$ \frac{\partial}{\partial Y}\mathcal{L} = 2LY - 2\Lambda Y = 0 \quad \Longrightarrow \quad (L - \Lambda)Y = 0 $$
This looks almost like a generalized eigenvalue problem, but $\Lambda$ is a diagonal matrix of unknown Lagrange multipliers.
Is there any way to solve such a problem?