Frobenius covariant

In matrix theory, the Frobenius covariants of a square matrix Template:Mvar are special polynomials of it, namely projection matrices Template:Math associated with the eigenvalues and eigenvectors of Template:Mvar.Template:Rp 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 Template:Math as a matrix polynomial, namely a linear combination of that function's values on the eigenvalues of Template:Mvar.

Formal definition

Let Template:Mvar be a diagonalizable matrix with eigenvalues λ₁, ..., λ_k.

The Frobenius covariant Template:Math, 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, Template:Math 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 Template:Mvar can be obtained from any eigendecomposition Template:Math, where Template:Mvar is non-singular and Template:Mvar is diagonal with Template:Math. The matrix Template:Mvar is defined up to multiplication on the right by a diagonal matrix. If Template:Mvar has no multiple eigenvalues, then let c_i be the Template:Mvarth right eigenvector of Template:Mvar, that is, the Template:Mvarth column of Template:Mvar; and let r_i be the Template:Mvarth left eigenvector of Template:Mvar, namely the Template:Mvarth row of Template:Mvar⁻¹. Then Template:Math. As a projection matrix, the Frobenius covariant satisfies the relation

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

which leads to Template:Math

Given that Template:Mvar and Template:Mvar are the right and left vectors satisfying Template:Math, the right and left eigenvectors of Template:Mvar may be written as Template:Math and Template:Math. 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 Template:Mvar.

If Template:Mvar has an eigenvalue λ_i appearing multiple times, then Template:Math, where the sum is over all rows and columns associated with the eigenvalue λ_i.Template:Rp

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, Template:Math.

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

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

with

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

Note Template:Math, as required.

Generalization

When a square matrix $A\in {ℂ}^{n\times n}$ has repeated eigenvalues (such as a defective matrix), the standard definition of Frobenius covariants based on Lagrange polynomials becomes undefined due to division by zero.

To generalize the projection matrices for these cases, one can use Hermite interpolating polynomials via the Chinese remainder theorem. Let ${\lambda }_1,\dots ,{\lambda }_k$ be the distinct eigenvalues of $A$, each with an algebraic multiplicity of $m_i+1$.

Using the node polynomial construction from Hermite interpolation, define:

$$f_i(\lambda )=u_i(\lambda )q_i(\lambda ),$$

where $q_i(\lambda )={\prod }_{r\ne i}(\lambda -{\lambda }_r{)}^{m_r+1}$. The polynomial $u_i(\lambda )$ is computed using the polynomial extended Euclidean algorithm to satisfy Bézout's identity:

$$u_i(\lambda )q_i(\lambda )+v_i(\lambda )(\lambda -{\lambda }_i{)}^{m_i+1}=1.$$

Evaluating this node polynomial at the matrix $A$ defines the generalized Frobenius covariant $A_i\in {ℂ}^{n\times n}$:

$$A_i=f_i(A).$$

The matrix $A_i$ remains a projection (i.e., $A_i^2=A_i$) that projects ${ℂ}^n$ onto the generalized eigenspace corresponding to ${\lambda }_i$ along the direct sum of the remaining generalized eigenspaces.