Compute the companion matrix for the polynomial 5+2x+3x²+x³:

Use a polynomial specification:

CompanionMatrix takes explicit polynomials:

CompanionMatrix takes symbolic input:

Give a list of coefficients and show its corresponding companion matrix:

CompanionMatrix uses a Right specification by default:

Instead locate the coefficients in the bottom row:

Check that these are transposes:

Locate the coefficients in the top row:

Locate the coefficients in the leftmost column:

Check that these are transposes:

Display the four cases:

CompanionMatrix[{a₀+a₁x+…+a_n-1 x^n-1+xⁿ,x}] and CompanionMatrix[{a₀,…,a_n-1}] are the same:

For , MatrixMinimalPolynomial[CompanionMatrix[c],x] is 's associated polynomial :

Then CharacteristicPolynomial[c,x] is times the minimal polynomial:

Compute the eigenvalues of a companion matrix:

These are the same as the roots of the corresponding polynomial:

All four possible layouts have the same eigenvalues:

The determinant of CompanionMatrix[{c₀,c₁,…}] is c₀:

Equivalently, it is the constant term in the associated polynomial :

As a consequence, the matrix is invertible as long as :

Compute a characteristic polynomial of a matrix:

Build the companion matrix corresponding to that polynomial:

This agrees with the Frobenius normal form:

For c==CompanionMatrix[[c₀,c₁,…}], the vectors , , , are the standard basis for :

Then is then

A coefficient list of two elements where the second is symbolic will be interpreted as a polynomial with the second element in the list being the variable:

Instead obtain the companion matrix of the polynomial with constant and linear terms corresponding to 3+x and x, respectively, and monic quadratic term: