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Concept
Companion matrix
Summary
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 . In this sense, the matrix and the polynomial are "companions".
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:
A is similar over F to , i.e. A can be conjugated to its companion matrix by matrices in GLn(F);
the characteristic polynomial coincides with the minimal polynomial of A , i.e. the minimal polynomial has degree n ;
the linear mapping makes a cyclic -module, having a basis of the form ; or equivalently as -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.
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.
Official source
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