Frobenius Covariant
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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 ?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 ?1, …, ?k.

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

${\displaystyle A_{i}\equiv \prod _{j=1 \atop j\neq i}^{k}{\frac {1}{\lambda _{i}-\lambda _{j}}}(A-\lambda _{j}I)~.}$

It is essentially the Lagrange polynomial with matrix argument. If the eigenvalue ?i is simple, then as an idempotent projection matrix to a one-dimensional subspace, Ai has a unit trace.

## Computing the covariants

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 = ?i. 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 = ciri.

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

## Example

Consider the two-by-two matrix:

${\displaystyle A={\begin{bmatrix}1&3\\4&2\end{bmatrix}}.}$

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

The corresponding eigen decomposition is

${\displaystyle A={\begin{bmatrix}3&1/7\\4&-1/7\end{bmatrix}}{\begin{bmatrix}5&0\\0&-2\end{bmatrix}}{\begin{bmatrix}3&1/7\\4&-1/7\end{bmatrix}}^{-1}={\begin{bmatrix}3&1/7\\4&-1/7\end{bmatrix}}{\begin{bmatrix}5&0\\0&-2\end{bmatrix}}{\begin{bmatrix}1/7&1/7\\4&-3\end{bmatrix}}.}$

Hence the Frobenius covariants, manifestly projections, are

${\displaystyle {\begin{array}{rl}A_{1}&=c_{1}r_{1}={\begin{bmatrix}3\\4\end{bmatrix}}{\begin{bmatrix}1/7&1/7\end{bmatrix}}={\begin{bmatrix}3/7&3/7\\4/7&4/7\end{bmatrix}}=A_{1}^{2}\\A_{2}&=c_{2}r_{2}={\begin{bmatrix}1/7\\-1/7\end{bmatrix}}{\begin{bmatrix}4&-3\end{bmatrix}}={\begin{bmatrix}4/7&-3/7\\-4/7&3/7\end{bmatrix}}=A_{2}^{2}~,\end{array}}}$

with

${\displaystyle A_{1}A_{2}=0,\qquad A_{1}+A_{2}=I~.}$

Note trA1=trA2=1, as required.

## References

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