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Frobenius covariant

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In matrix theory, the Frobenius covariants of a square matrix A are special polynomials of it, namely projection matrices Fi(A) associated with the eigenvalues and eigenvectors of A.[1]:pp.403,437–8 They are named after the mathematician Ferdinand Frobenius.

Each covariant is a projection on the eigenspace associated with the eigenvalue λi. 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.

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Formal definition

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

The Frobenius covariant Fi(A), for i = 1,..., k, is the matrix

It is essentially the Lagrange polynomial with matrix argument. If the eigenvalue λi is simple, then as an idempotent projection matrix to a one-dimensional subspace, Fi(A) has a unit trace.

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Computing the covariants

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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 Di,i = λi. The matrix S is defined up to multiplication on the right by a diagonal matrix. If A has no multiple eigenvalues, then let ci be the ith right eigenvector of A, that is, the ith column of S; and let ri be the ith left eigenvector of A, namely the ith row of S−1. Then Fi(A) = ci ri. As a projection matrix, the Frobenius covariant satisfies the relation

which leads to ri cj = δij.

Given that v and w are the right and left vectors satisfying w Fi(A) v ≠ 0, the right and left eigenvectors of A may be written as ci = Nci Fi(A) v and ri = Nri w Fi(A). The orthonormality of the eigenvectors gives one constraint for the normalization coefficients. The remaining freedom is related to the choice of representation for the matrix S.

If A has an eigenvalue λi appearing multiple times, then Fi(A) = Σj cj rj, where the sum is over all rows and columns associated with the eigenvalue λi.[1]:p.521

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Example

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Consider the two-by-two matrix:

This matrix has two eigenvalues, 5 and −2, which can be found by solving the characteristic equation. By virtue of the Cayley–Hamilton theorem, (A − 5)(A + 2) = 0.

The corresponding eigen decomposition is

Hence the Frobenius covariants, manifestly projections, are

with

Note trF1(A) = trF2 (A)= 1, as required.

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References

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