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

Formal definition

Let A be a diagonalizable matrix with eigenvalues λ1, ..., λk. The Frobenius covariant Ai , 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, Ai has a unit trace.

Computing the covariants

The Frobenius covariants of a matrix A can be obtained from any eigendecomposition , where S is non-singular and D is diagonal with Di,i = λi . If A has no multiple eigenvalues, then let ci be the ith right eigenvector of A, that is, the ith column of S; and let ri be the ith left eigenvector of A, namely the ith row of S−1. Then . If A 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 (A − 5)(A + 2) = 0 . The corresponding eigen decomposition is Hence the Frobenius covariants, manifestly projections, are with Note , as required.