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.

Formal definition

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

The Frobenius covariant A_i, for i = 1,..., k, is the matrix

[math]\displaystyle{ A_i \equiv \prod_{j=1 \atop j \ne i}^k \frac{1}{\lambda_i-\lambda_j} (A - \lambda_j I)~. }[/math]

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.

Computing the covariants

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

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 right eigenvector of A, that is, the ith column of S; and let r_i be the ith left eigenvector of A, namely the ith row of S⁻¹. Then A_i = c_i r_i.

If A has an eigenvalue λ_i appearing multiple times, then A_i = Σ_j c_j r_j, where the sum is over all rows and columns associated with the eigenvalue λ_i.

Example

Consider the two-by-two matrix:

[math]\displaystyle{ A = \begin{bmatrix} 1 & 3 \\ 4 & 2 \end{bmatrix}. }[/math]

This matrix has two eigenvalues, 5 and −2; hence (A − 5)(A + 2) = 0.

The corresponding eigen decomposition is

[math]\displaystyle{ A = \begin{bmatrix} 3 & 1/7 \\ 4 & -1/7 \end{bmatrix} \begin{bmatrix} 5 & 0 \\ 0 & -2 \end{bmatrix} \begin{bmatrix} 3 & 1/7 \\ 4 & -1/7 \end{bmatrix}^{-1} = \begin{bmatrix} 3 & 1/7 \\ 4 & -1/7 \end{bmatrix} \begin{bmatrix} 5 & 0 \\ 0 & -2 \end{bmatrix} \begin{bmatrix} 1/7 & 1/7 \\ 4 & -3 \end{bmatrix}. }[/math]

Hence the Frobenius covariants, manifestly projections, are

[math]\displaystyle{ \begin{array}{rl} A_1 &= c_1 r_1 = \begin{bmatrix} 3 \\ 4 \end{bmatrix} \begin{bmatrix} 1/7 & 1/7 \end{bmatrix} = \begin{bmatrix} 3/7 & 3/7 \\ 4/7 & 4/7 \end{bmatrix} = A_1^2\\ A_2 &= c_2 r_2 = \begin{bmatrix} 1/7 \\ -1/7 \end{bmatrix} \begin{bmatrix} 4 & -3 \end{bmatrix} = \begin{bmatrix} 4/7 & -3/7 \\ -4/7 & 3/7 \end{bmatrix}=A_2^2 ~, \end{array} }[/math]

with

[math]\displaystyle{ A_1 A_2 = 0 , \qquad A_1 + A_2 = I ~. }[/math]

Note tr A₁ = tr A₂ = 1, as required.