What is the meaning of Eigenvalue?

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

Source: wiktionary.org