Frobenius covariant


In matrix theory, the Frobenius covariants of a square matrix are special polynomials of it, namely projection matrices Ai 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.
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, …, λk.
The Frobenius covariant, for i = 1,…, k, is the matrix
It is essentially the Lagrange polynomial with matrix argument. If the eigenvalue λi 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.
If has no multiple eigenvalues, then let ci be the th right eigenvector of, that is, the th column of ; and let ri be the th left eigenvector of, namely the th row of −1. Then.
If has an eigenvalue λi appear multiple times, then, where the sum is over all rows and columns associated with the eigenvalue λi.

Example

Consider the two-by-two matrix:
This matrix has two eigenvalues, 5 and −2; hence =0.
The corresponding eigen decomposition is
Hence the Frobenius covariants, manifestly projections, are
with
Note, as required.