The characteristic polynomial of a square matrix is the polynomial formed with roots at the eigenvalues of the matrix . The characteristic polynomial of square matrix is defined as
which expands out to an degree polynomial in . The roots of (where ) are the eigenvalues of .
Factoring the Characteristic Polynomial
Special Cases
Matrix
For a matrix it's characteristic polynomial can be calculated as follows:
For triangular matrices of size their characteristic polynomials can be calculated as follows:
This means that for triangular matrices their eigenvalues are just their diagonal entries.
Example
Find the characteristic polynomial of the matrix