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.

References