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).
Any matrix A with entries in a field F has characteristic polynomial
The following statements are equivalent: If the above hold, one says that A is non-derogatory.
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.
Indeed, a reasonably hard computation shows that the transpose
Thus, its diagonalizing change of basis matrix is
, and taking the transpose of both sides gives
: 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
Rather, the Jordan canonical form of
contains one Jordan block for each distinct root; if the multiplicity of the root is m, then the block is an m × m matrix with
in this case, V becomes a confluent Vandermonde matrix.
[2] A linear recursive sequence defined by
, whose transpose companion matrix
Setting the initial values of the sequence equal to this vector produces a geometric sequence
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.
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
is the transpose companion matrix for the characteristic polynomial
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.
, 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.
is unchanged, but as above, it can be diagonalized by matrices with entries in
The explicit formula for the eigenvectors (the scaled column vectors of the inverse Vandermonde matrix
are the coefficients of the scaled Lagrange polynomial