Find the characteristic polynomial of a matrix with integer entries:

Visualize the polynomial:

Find the characteristic polynomial in of the symbolic matrix :

Compare with a direct computation:

Compute the characteristic polynomials of the identity matrix and zero matrix:

Find the characteristic polynomial of the matrix and compare the behavior for , and :

Examining the roots, there is a root at independent of :

For the root at is repeated:

For there are three distinct real roots:

And for , is the only real root, with the other two roots a complex conjugate pair:

Visualize the three polynomials, zooming in on the "bounce" of the plot at the double root :

Compute the determinant of a matrix as the constant term in its characteristic polynomial:

Substitute in :

This result is also the product of the roots of the characteristic polynomial:

Compare with a direct computation using Det :

Compute the trace of a matrix as the coefficient of the subleading power term in the characteristic polynomial:

Extract the coefficient of , where is the height or width of the matrix:

This result is also the sum of the roots of the characteristic polynomial:

Compare with a direct computation using Tr :

Find the eigenvalues of a matrix as the roots of the characteristic polynomial:

Compare with a direct computation using Eigenvalues :

Use the characteristic polynomial to find the eigenvalues and eigenvectors of the matrices and TemplateBox[{a}, Transpose]:

The two matrices have the same characteristic polynomial:

Thus, they will both have the same eigenvalues, which are the roots of the polynomial:

The eigenvectors are given by the null space of :

Eigensystem gives the same result, though it sorts eigenvalues by absolute value:

While TemplateBox[{a}, Transpose] has the same eigenvalues as , it has different eigenvectors:

Visualize the two sets of eigenvectors:

Find the generalized eigensystem of with respect to as the roots of the characteristic polynomial:

The roots of the generalized characteristic polynomial are the generalized eigenvalues:

The generalized eigenvectors are given by the null space of :

Compare with a direct computation using Eigensystem :

The characteristic polynomial is equivalent to Det [m-id x]:

The generalized characteristic polynomial is equivalent to Det [m-a x]:

A matrix is a root of its characteristic polynomial (Cayley–Hamilton theorem [more...]):

where are the eigenvalues is equivalent to the characteristic polynomial:

The sum of the roots of the characteristic polynomial is the trace (Tr ) of the matrix:

Similarly, the product of the roots is the determinant (Det ):

A matrix and its transpose have the same characteristic polynomial:

All triangular matrices with a common diagonal have the same characteristic polynomial:

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

MatrixMinimalPolynomial [m,λ] divides CharacteristicPolynomial [m,λ] with a (possibly constant) polynomial quotient:

The minimal polynomial and the characteristic polynomial have the same distinct roots:

In particular, this means divides raised to the power: