Frobenius covariant

In matrix theory, the Frobenius covariants of a square matrix {{mvar|A}} are special polynomials of it, namely projection matrices A_i associated with the eigenvalues and eigenvectors of {{mvar|A}}.Roger A. Horn and Charles R. Johnson (1991), Topics in Matrix Analysis. Cambridge University Press, {{ISBN|978-0-521-46713-1}}{{rp|pp.403,437–8}} They are named after the mathematician Ferdinand Frobenius.

Each covariant is a projection on the eigenspace associated with the eigenvalue {{math|λ_i}}.

Frobenius covariants are the coefficients of Sylvester's formula, which expresses a function of a matrix {{math|f(A)}} as a matrix polynomial, namely a linear combination

of that function's values on the eigenvalues of {{mvar|A}}.

Formal definition

Let {{mvar|A}} be a diagonalizable matrix with eigenvalues λ₁, ..., λ_k.

The Frobenius covariant {{math|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. If the eigenvalue λ_i is simple, then as an idempotent projection matrix to a one-dimensional subspace, {{math|A_i}} has a unit trace.

Computing the covariants

File:GeorgFrobenius.jpg (1849–1917), German mathematician. His main interests were elliptic functions differential equations, and later group theory.]]

The Frobenius covariants of a matrix {{mvar|A}} can be obtained from any eigendecomposition {{math|A {{=}} SDS⁻¹}}, where {{mvar|S}} is non-singular and {{mvar|D}} is diagonal with {{math|D_i,i {{=}} λ_i}}.

If {{mvar|A}} has no multiple eigenvalues, then let c_i be the {{mvar|i}}th right eigenvector of {{mvar|A}}, that is, the {{mvar|i}}th column of {{mvar|S}}; and let r_i be the {{mvar|i}}th left eigenvector of {{mvar|A}}, namely the {{mvar|i}}th row of {{mvar|S}}⁻¹. Then {{math|A_i {{=}} c_i r_i}}.

If {{mvar|A}} has an eigenvalue λ_i appearing multiple times, then {{math|A_i {{=}} Σ_j c_j r_j}}, where the sum is over all rows and columns associated with the eigenvalue λ_i.{{rp|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 {{math| (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 {{math|tr{{nnbsp}}A₁ {{=}} tr{{nnbsp}}A₂ {{=}} 1}}, as required.

References

Category:Matrix theory