eigenvector

An eigenvector is a (nonzero) vector that has its direction unchanged by a given linear transformation. More precisely, an eigenvector $\overrightarrow{v}$ of a linear transformation $T$ is scaled by a constant factor $\lambda$ called the eigenvalue, i.e. $T(\overrightarrow{v})=\lambda \overrightarrow{v}$.

Because linear transformations over a finite-dimensional vector space can be represented using an $n\times n$ matrix, we can express the above as a matrix equation, $A\overrightarrow{v}=\lambda \overrightarrow{v}$

To solve for an eigenvector, we can equivalently state the above equation as

where $I$ is the identity matrix and $\overrightarrow{0}$ is the zero vector. This equation has a nonzero solution if and only if the determinant of $(A-\lambda I)$ is zero; thus, the eigenvalues are the values of $\lambda$ which make the determinant zero. This is a polynomial of $\lambda$, called the Characteristic Polynomial of $A$.