Frobenius covariant

In matrix theory, the Frobenius covariants of a square matrix A are special polynomials of it, namely projection matrices A_i 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 λ_i. 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.

Let A be a diagonalizable matrix with eigenvalues λ₁, …, λ_k.

The Frobenius covariant A_i, 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, A_i 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 D_i,i = λ_i. If A has no multiple eigenvalues, then let c_i be the ith left eigenvector of A, that is, the ith column of S; and let r_i be the ith right eigenvector of A, namely the ith row of S⁻¹. Then A_i = c_ir_i.

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