Frobenius covariant

In matrix theory, the Frobenius covariants of a square matrix A are matrices A_i associated with the eigenvalues and eigenvectors of A.^[1] Each covariant is a projection on the eigenspace associated with λ_i.

Frobenius covariants are the coefficients of Sylvester's formula, that expresses a function of a matrix f(A) as a linear combination of its values on the eigenvalues of A. They are named after the mathematician Ferdinand Frobenius.

Contents 1 Formal definition 2 Computing the covariants 3 Example 4 References

Formal definition

Let A be a diagonalizable matrix with k distinct eigenvalues, λ₁, …, λ_k. The Frobenius covariant A_i, for i = 1,…, k, is the matrix

Computing the covariants

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.

If A has multiple eigenvalues then A_i = Σ_j c_jr_j, where the sum is over all rows and columns associated with the eigenvalue λ_i.^[1]:p.521

Example

Consider the two-by-two matrix:

This matrix has two eigenvalues, 5 and −2. The corresponding eigen decomposition is

Hence the Frobenius covariants are

References