In linear algebra, the Frobenius companion matrix of the monic polynomial is the square matrix defined as

Some authors use the transpose, which is more convenient for some purposes such as linear recurrence relations (see below).

, while the characteristic polynomial as well as the minimal polynomial. are "companions".

Similarity to companion matrix

Any matrix with entries in a field . These matrices are related as follows.

The following statements are equivalent:

• A is similar over F, i.e. A can be conjugated to its companion matrix by matrices in GLn(F);

• coincides with the minimal polynomial of A, i.e. the minimal polynomial has degree n;

• a cyclic-modules.

If the above hold, one says that A is non-derogatory.

Not every square matrix is similar to a companion matrix, but every square matrix is similar to a block diagonal matrix made of companion matrices. If we also demand that the polynomial of each diagonal block divides the next one, they are uniquely determined by A, and this gives the rational canonical form of A.

Diagonalizability

are the eigenvalues.

If there are n is diagonalizable, where D is the diagonal matrix and V is the Vandermonde matrix corresponding to the . Thus, its diagonalizing change of basis: they are the column vectors of the inverse Vandermonde matrix, with coordinates equal to the coefficients of the Lagrange polynomials have simpler coefficients.

is not diagonalizable. Rather, the Jordan canonical form contains one Jordan block for each distinct root; if the multiplicity of the root is m, then the block is an m × m on the diagonal and 1 in the entries just above the diagonal. In this case, V becomes a confluent Vandermonde matrix.

Linear recursive sequences

A linear recursive sequence which satisfies the recurrence. In the case of n can be written as a linear combination of such geometric solutions, and the eigenvalues of largest complex norm give an asymptotic approximation.

From linear ODE to first-order linear ODE system

Similarly to the above case of linear recursions, consider a homogeneous linear ODE of order n may be also functions, not just constants.

is diagonalizable, then a diagonalizing change of basis will transform this into a decoupled system equivalent to one scalar homogeneous first-order linear ODE in each coordinate.

.

Again, a diagonalizing change of basis will transform this into a decoupled system of scalar inhomogeneous first-order linear ODEs.

Cyclic shift matrix

, when the eigenvalues are the complex roots of unity, the companion matrix and its transpose both reduce to Sylvester's cyclic shift matrix, a circulant matrix.

Multiplication map on a simple field extension

with coefficients in a field is irreducible in the polynomial ring. Then adjoining a root produces a field extension-linear multiplication mapping has an n × n Assuming this extension is separable has characteristic zero or is a finite field and it has splitting field; rather, we must extend.

and the Vandermonde matrix V. The explicit formula for the eigenvectors (the scaled column vectors of the inverse Vandermonde matrix

Theoretical complexity: calculation by fast matrix multiplication

It is possible to calculate the companion matrix in a fast way with the use of fast matrix multiplication algorithms. Respective algorithms is given by

See also

• Frobenius endomorphism

• Cayley–Hamilton theorem

• Krylov subspace

Notes