In this section, the field of scalars F will be ℂ. The following is the definition of eigenvalues and eigenvectors.
Definition A.7.1 Let M be an n × n matrix and let x ∈ ℂn be a nonzero vector for which
The set of all eigenvalues of an n×n matrix M, is denoted by σ
The following corollary of Theorem A.6.5 tells how to obtain eigenvalues and eigenvectors.
Corollary A.7.2 Let M be an n × n matrix, the entries in a field of scalars F, for us ℂ. Then λ is an eigenvalue of M if and only if det
Note that the above corollary is valid if M − λI is replaced with λI − M.
If you have an eigenvalue λ ∈ F, for us F = ℂ, then to get the eigenvectors for this λ, you find nonzero solutions x to