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 $k$ distinct eigenvalues, λ₁, …, λ_k.

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

A_i \equiv \prod_{j=1 \atop j \ne i}^k \frac{1}{\lambda_i-\lambda_j} (A - \lambda_j I)~.

It is essentially the Lagrange polynomial with matrix argument. As an idempotent projection matrix to a one-dimensional subspace, it 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 $D i, i = λ i$ . If $A$ has no multiple eigenvalues, then let c_i be the $i$ th left eigenvector of $A$ , that is, the $i$ th column of $S$ ; and let r_i be the $i$ th right eigenvector of $A$ , namely the $i$ th row of $S$ ⁻¹. Then $A i = c i r i$ .

If $A$ has multiple eigenvalues, 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:

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

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

\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

A_1 A_2 =0 , \qquad A_1+A_2= I ~.

Note $tr A 1 =tr A 2 =1$ , as required.

References

1 2 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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