Eigenvalue and eigenvector

A vector $\mathbf{v} \in \mathbb{R}^d$ is an eigenvector of a matrix $\mathbf{X} \in \mathbb{R}^{d \times d}$, if there exists a scalar $\lambda$ such that $$\mathbf{Xv} = \lambda \mathbf{v}$$ where the scalar $\lambda$ is called the eigenvalue associated with $\mathbf{v}$.