Eigenvalues and Eigenvectors
In abstract mathematics, a more general definition is given :
Let $$$$V$$$$ be any vector space, let $$$$\textbf{x}$$$$ be a vector in that vector space, and let $$$$T$$$$ be a linear transformation mapping $$$$V$$$$ into $$$$V$$$$. Then $$$$\textbf{x}$$$$ is an eigenvector of $$$$T$$$$ with eigenvalue $$$$\lambda$$$$ if the following equation holds:
$$$$ T\textbf{x} = \lambda \textbf{x} $$$$
Usually, the multiplication of a vector $$$$\textbf{x}$$$$ by a square matrix $$$$T$$$$ changes both the magnitude and the direction of the vector it acts on—but in the special case where it changes only the scale of the vector and leaves the direction unchanged, or switches the vector to the opposite direction, that vector is called an eigenvector of that matrix.
