# Generalization of eigendecomposition problem

Let $$A\in \mathbb{R}^{n\times n}$$ and $$v \in \mathbb{R}^n$$. We recognize $$Av=\lambda v$$ for some scalar $$\lambda$$ as an eigendecomposition problem. Suppose $$\mu \in \mathbb{R}^n$$, and let $$\odot$$ denotes the Hadamard product. Does this system $$Av = \lambda v+ \mu \odot v$$ correspond to any generalization of eigendecomposition? Can we solve for $$\lambda, v$$?

$$B v = \lambda v\,$$
with $$B = A - M$$, $$M = \textrm{diag}(\mu)$$.