For an square matrix , its eigenvectors are non-zero vectors such that results in scalar multiple of . In other words, can scale, reverse the direction of, or rotate , but it will not otherwise transform it. can have at most linearly independent eigenvectors.
For each eigenvector , there is a corresponding characteristic (possibly complex-valued) scalar called an eigenvalue.
The eigenvalues of a square matrix can be discovered by calculating the determinant and solving for . The eigenvectors can then be discovered by solving for each . (See Eigenvalue decomposition for a square matrix.)
Interpretation of eigenvalues
For any eigenvalue :
The scaling factor is .
If has negative sign, the direction of the eigenvector is reversed.
Complex eigenvalues always appear in pairs, as complex conjugates. Such a pair of eigenvalues corresponds to a rotation in a 2-dimensional (sub)space of the -dimensional space. The angle of rotation in this subspace is given by