Eigenvalues and eigenvectors
with ; the admissible are exactly the roots of the characteristic polynomial .
Let (for a real or complex vector space ) be a linear map. Then, a vector with is an eigenvector of if there exists a scalar (or ) such that
The scalar is called the eigenvalue corresponding to the eigenvector .
If or , and is given in terms of a matrix , then
and for a given , this holds for some vector if and only if . This is called the characteristic equation of the matrix .
Furthermore, the polynomial is called the characteristic polynomial of degree of the matrix .
Examples
For , representing a rotation of , the characteristic polynomial is , so the eigenvalues are and , with eigenvectors and respectively.
For over , : the single eigenvalue has multiplicity , yet the eigenvectors are only , a single line.
Trace and determinant
Writing , the leading coefficients are and . Comparing with the roots :
so an matrix has eigenvalues counted with multiplicity, and real matrices have eigenvalues that are either real or come in complex conjugate pairs.
Related
Stated in
- Definition 4.3 (Eigenvector and Eigenvalue)§4.1 Introduction