Frobenius covariant explained

In matrix theory, the Frobenius covariants of a square matrix are special polynomials of it, namely projection matrices A_i associated with the eigenvalues and eigenvectors of .^[1] 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, ..., λ_k.

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

A _{i} \equiv \prod ^{k} _{j =1 \atop j \ne i} (1)/(λ _{i-λ} _{j}) (A - λ _{j} I)~. 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, has a unit trace.

See also: Resolvent formalism.

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 c_i be the th right eigenvector of, that is, the th column of ; and let r_i be the th left eigenvector of, namely the th row of ⁻¹. Then .

If has an eigenvalue λ_i appearing multiple times, then, where the sum is over all rows and columns associated with the eigenvalue λ_i.^[1]

Example

Consider the two-by-two matrix:

A = \begin{bmatrix} 1 & 3 \ 4 & 2 \end{bmatrix}. This matrix has two eigenvalues, 5 and −2; hence .

The corresponding eigen decomposition is

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}. Hence the Frobenius covariants, manifestly projections, are \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 ^{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} withA _{1} A _{2} = 0 , A _{1} + A _{2} = I ~. Note, as required.

Notes and References

1. Roger A. Horn and Charles R. Johnson (1991), Topics in Matrix Analysis. Cambridge University Press,