Frobenius covariant

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

Formal definition

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

The Frobenius covariant Ai, 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, Ai has a unit trace.

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 Di,i = λi. 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 Ai = ci ri.

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

Example

Consider the two-by-two matrix:

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

The corresponding eigen decomposition is

Hence the Frobenius covariants, manifestly projections, are

with

Note trA1=trA2=1, as required.

References

  1. Roger A. Horn and Charles R. Johnson (1991), Topics in Matrix Analysis. Cambridge University Press, ISBN 978-0-521-46713-1
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