Frobenius covariant

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.

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.

If the eigenvalue λi is simple, then as an idempotent projection matrix to a one-dimensional subspace, Ai has a unit trace.

The Frobenius covariants of a matrix A can be obtained from any eigendecomposition A = SDS−1, where S is non-singular and D is diagonal with Di,i = λi.

If A has an eigenvalue λi appearing multiple times, then Ai = Σj cj rj, where the sum is over all rows and columns associated with the eigenvalue λi.

Ferdinand Georg Frobenius (1849–1917), German mathematician. His main interests were elliptic functions differential equations , and later group theory .