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 $\mathbf{v}$ .

Notes

intuitively: the eigenvector does not change direction upon application of linear transformation $\mathbf{X}$ , and merely is scaled by $\lambda$