In linear algebra, the Frobenius companion matrix of the monic polynomial

is the square matrix defined as

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

is defined from the coefficients of , while the characteristic polynomial as well as the minimal polynomial of are equal to .[1] In this sense, the matrix and the polynomial are "companions".

Similarity to companion matrix

Any matrix A with entries in a field F has characteristic polynomial , which in turn has companion matrix . These matrices are related as follows.

The following statements are equivalent:

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

The roots of the characteristic polynomial are the eigenvalues of . If there are n distinct eigenvalues , then is diagonalizable as , where D is the diagonal matrix and V is the Vandermonde matrix corresponding to the λ's:

Indeed, an easy computation shows that the transpose has eigenvectors with , which follows from . Thus, its diagonalizing change of basis matrix is , meaning , and taking the transpose of both sides gives .

We can read the eigenvectors of with from the equation : they are the column vectors of the inverse Vandermonde matrix . This matrix is known explicitly, giving the eignevectors , with coordinates equal to the coefficients of the Lagrange polynomials

Alternatively, the scaled eigenvectors have simpler coefficients.

If has multiple roots, then is not diagonalizable. Rather, the Jordan canonical form of contains one diagonal block for each distinct root, an m × m block with on the diagonal if the root has multiplicity m.

Linear recursive sequences

A linear recursive sequence defined by for has the characteristic polynomial , whose transpose companion matrix generates the sequence:

The vector is an eigenvector of this matrix, where the eigenvalue is a root of . Setting the initial values of the sequence equal to this vector produces a geometric sequence which satisfies the recurrence. In the case of n distinct eigenvalues, an arbitrary solution 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 for the scalar function :

This can be equivalently described as a coupled system of homogeneous linear ODE of order 1 for the vector function :
where is the transpose companion matrix for the characteristic polynomial
Here the coefficients may be also functions, not just constants.

If 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.

An inhomogeneous equation

is equivalent to the system:
with the inhomogeneity term .

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

Cyclic shift matrix

In the case of , 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

Consider a polynomial with coefficients in a field , and suppose is irreducible in the polynomial ring . Then adjoining a root of produces a field extension , which is also a vector space over with standard basis . Then the -linear multiplication mapping

defined by

has an n × n matrix with respect to the standard basis. Since and , this is the companion matrix of :

Assuming this extension is separable (for example if has characteristic zero or is a finite field), has distinct roots with , so that
and it has splitting field . Now is not diagonalizable over ; rather, we must extend it to an -linear map on , a vector space over with standard basis , containing vectors . The extended mapping is defined by .

The matrix is unchanged, but as above, it can be diagonalized by matrices with entries in :

for the diagonal matrix and the Vandermonde matrix V corresponding to . The explicit formula for the eigenvectors (the scaled column vectors of the inverse Vandermonde matrix ) can be written as:
where are the coefficients of the scaled Lagrange polynomial

See also

Notes

  1. ^ Horn, Roger A.; Charles R. Johnson (1985). Matrix Analysis. Cambridge, UK: Cambridge University Press. pp. 146–147. ISBN 0-521-30586-1. Retrieved 2010-02-10.