# 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.[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 *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 *λ*_{1}, …, *λ*_{k}.

The Frobenius covariant *A*_{i}, for *i* = 1,…, *k*, is the matrix

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

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} appear 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}.[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 tr*A*_{1}=tr*A*_{2}=1, as required.

## References

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