What is the meaning of Eigenvalue?

A scalar ${\displaystyle \lambda }$, such that there exists a non-zero vector ${\displaystyle \mathrm {x} }$ (a corresponding eigenvector) for which the image of ${\displaystyle \mathrm {x} }$ under a given linear transformation ${\displaystyle A}$ is equal to the image of ${\displaystyle \mathrm {x} }$ under multiplication by ${\displaystyle \lambda }$; i.e. ${\displaystyle A\mathrm {x} =\lambda \mathrm {x} }$.

Source: wiktionary.org