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 eigenvalues λ₁, …, λ_k.

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

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 λ_i is simple, then as an idempotent projection matrix to a one-dimensional subspace, $A i$ 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 right eigenvector of $A$ , that is, the $i$ th column of $S$ ; and let r_i be the $i$ th left eigenvector of $A$ , namely the $i$ th row of $S$ ⁻¹. Then $A i = c i r i$ .

If $A$ has an eigenvalue λ_i appear multiple times, 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.

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References

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