This is documentation for Mathematica 8, which was
based on an earlier version of the Wolfram Language.

# CharacteristicPolynomial

 CharacteristicPolynomialgives the characteristic polynomial for the matrix m. CharacteristicPolynomialgives the generalized characteristic polynomial with respect to a.
• m must be a square matrix.
• It can contain numeric or symbolic entries.
• is essentially equivalent to Det where id is the identity matrix of appropriate size. »
 Out[1]=
 Out[2]=
 Scope   (2)
Use exact arithmetic to find the characteristic polynomial:
Use machine arithmetic:
Use 20-digit precision arithmetic:
The characteristic polynomial of a complex matrix:
The generalized characteristic polynomial :
 Applications   (1)
Find the eigenvalues of a matrix as the roots of the characteristic polynomial:
The characteristic polynomial is equivalent to Det:
The generalized characteristic polynomial is equivalent to Det:
A matrix is a root of its characteristic polynomial (Cayley-Hamilton theorem []):
Evaluate the polynomial at m with matrix arithmetic:
Use the more efficient Horner's method to evaluate the polynomial:
where are the eigenvalues is equivalent to the characteristic polynomial:
If is a monic polynomial, then the characteristic polynomial of its companion matrix is:
Form the companion matrix: