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 appearing 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 .
The corresponding eigen decomposition is
Hence the Frobenius covariants, manifestly projections, are
with
Note , as required.