Frobenius covariant

In matrix theory, the Frobenius covariants of a square matrix are special polynomials of it, namely projection matrices associated with the eigenvalues and eigenvectors of . They are named after the mathematician Ferdinand Frobenius.
Each covariant is a projection on the eigenspace associated with the eigenvalue λ_i.
Frobenius covariants are the coefficients of Sylvester's formula, which expresses a function of a matrix as a matrix polynomial, namely a linear combination
of that function's values on the eigenvalues of.

Formal definition

Let be a diagonalizable matrix with eigenvalues λ₁,..., λ_k.
The Frobenius covariant, for i = 1,..., k, is the matrix
It is essentially the Lagrange polynomial with matrix argument. If the eigenvalue λ_i is simple, then as an idempotent projection matrix to a one-dimensional subspace, has a unit trace.

Computing the covariants

The Frobenius covariants of a matrix can be obtained from any eigendecomposition, where is non-singular and is diagonal with.
The matrix is defined up to multiplication on the right by a diagonal matrix.
If has no multiple eigenvalues, then let c_i be the th right eigenvector of, that is, the th column of ; and let r_i be the th left eigenvector of, namely the th row of ⁻¹. Then.
As a projection matrix, the Frobenius covariant satisfies the relation
which leads to
Given that
and are the right and left vectors satisfying
the right and left eigenvectors of may be written as
and
.
The orthonormality of the eigenvectors gives one constraint for the normalization coefficients. The remaining freedom is related to the choice of representation for the matrix.
If has an eigenvalue λ_i appearing multiple times, then, where the sum is over all rows and columns associated with the eigenvalue λ_i.

Example

Consider the two-by-two matrix:
This matrix has two eigenvalues, 5 and −2, which can be found by solving the characteristic equation. By virtue of the Cayley–Hamilton theorem,.
The corresponding eigen decomposition is
Hence the Frobenius covariants, manifestly projections, are
with
Note, as required.