Frobenius covariant

In matrix theory, the Frobenius covariants of a square matrix A are special polynomials of it, namely projection matrices F_i(A) associated with the eigenvalues and eigenvectors of A. 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 λ₁, ..., λ_k.

The Frobenius covariant F_i(A), for i = 1,..., k, is the matrix

$F_i(A)\equiv \prod _{\genfrac{}{}{0ex}{}{j=1}{j\ne i}}^k\frac{1}{{\lambda }_i-{\lambda }_j}(A-{\lambda }_{j}I).$

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.

See also: Resolvent formalism

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. 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 F_i(A) = c_i r_i. As a projection matrix, the Frobenius covariant satisfies the relation

$F_i(A)F_j(A)={\delta }_{ij}{F}_i(A),$

which leads to r_i c_j = δ_ij.

Given that v and w are the right and left vectors satisfying w F_i(A) v Template:≠ 0, the right and left eigenvectors of A may be written as c_i = N_ci F_i(A) v and r_i = N_ri w F_i(A). 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 F_i(A) = Σ_j c_j r_j, where the sum is over all rows and columns associated with the eigenvalue λ_i.

Example

Consider the two-by-two matrix:

$A=\left[\begin{array}{cc}1& 3\\ 4& 2\end{array}\right].$

This matrix has two eigenvalues, 5 and −2, which can be found by solving the characteristic equation. By virtue of the Cayley–Hamilton theorem, (A − 5)(A + 2) = 0.

The corresponding eigen decomposition is

$A=\left[\begin{array}{cc}3& 1/7\\ 4& -1/7\end{array}\right]\left[\begin{array}{cc}5& 0\\ 0& -2\end{array}\right]{\left[\begin{array}{cc}3& 1/7\\ 4& -1/7\end{array}\right]}^{-1}=\left[\begin{array}{cc}3& 1/7\\ 4& -1/7\end{array}\right]\left[\begin{array}{cc}5& 0\\ 0& -2\end{array}\right]\left[\begin{array}{cc}1/7& 1/7\\ 4& -3\end{array}\right].$

Hence the Frobenius covariants, manifestly projections, are

$\begin{array}{rl}{F}_1(A)& =c_1{r}_1=\left[\begin{array}{c}3\\ 4\end{array}\right]\left[\begin{array}{cc}1/7& 1/7\end{array}\right]=\left[\begin{array}{cc}3/7& 3/7\\ 4/7& 4/7\end{array}\right]=F_1^2(A)\\ F_2(A)& =c_2{r}_2=\left[\begin{array}{c}1/7\\ -1/7\end{array}\right]\left[\begin{array}{cc}4& -3\end{array}\right]=\left[\begin{array}{cc}4/7& -3/7\\ -4/7& 3/7\end{array}\right]=F_2^2(A),\end{array}$

with

$F_1(A)F_2(A)=0,F_1(A)+F_2(A)=I.$

Note tr F₁(A) = tr F₂ (A)= 1, as required.