Frobenius covariant

Formal definition

Let A be a diagonalizable matrix with eigenvalues λ1, ..., λ**k.

The Frobenius covariant F**i(A), for i = 1,..., k, is the matrix : F_i (A) \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. If the eigenvalue λ**i is simple, then as an idempotent projection matrix to a one-dimensional subspace, F**i(A) has a unit trace.

Computing the covariants

The Frobenius covariants of a matrix A can be obtained from any eigendecomposition , where S is non-singular and D is diagonal with . The matrix S is defined up to multiplication on the right by a diagonal matrix. 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 . As a projection matrix, the Frobenius covariant satisfies the relation : F_i (A)F_j (A) = \delta_{ij}F_i (A), which leads to

Given that v and w are the right and left vectors satisfying w F**i(A) v 0, the right and left eigenvectors of A may be written as and . 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 λ**i appearing multiple times, then , where the sum is over all rows and columns associated with the eigenvalue λ**i.

Example

Consider the two-by-two matrix: : A = \begin{bmatrix} 1 & 3 \ 4 & 2 \end{bmatrix}. This matrix has two eigenvalues, 5 and −2, which can be found by solving the characteristic equation. By virtue of the Cayley–Hamilton theorem, .

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} F_1(A) &= 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} = F_1^2(A)\ F_2(A) &= 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}=F_2^2(A) ~, \end{array} with :F_1(A) F_2(A) = 0 , \qquad F_1(A) + F_2(A) = I ~. Note , as required.

References

References

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