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BUILT-IN MATHEMATICA SYMBOL

CharacteristicPolynomial

gives the characteristic polynomial for the matrix m.

gives the generalized characteristic polynomial with respect to a.

is essentially equivalent to Det where id is the identity matrix of appropriate size. »

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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

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:

is a monic polynomial, then the characteristic polynomial of its companion matrix is

Form the companion matrix: