In matrix theory, the Frobenius covariants of a square matrix are special polynomials of it, namely projection matrices associated with the eigenvalues and eigenvectors of .Roger A. Horn and Charles R. Johnson (1991), Topics in Matrix Analysis. Cambridge University Press, They are named after the mathematician Ferdinand Frobenius.
Each covariant is a projection on the eigenspace associated with the eigenvalue . Frobenius covariants are the coefficients of Sylvester's formula, which expresses a function of a matrix as a matrix polynomial, namely a linear combination of that function's values on the eigenvalues of .
Formal definition Let be a diagonalizable matrix with eigenvalues .
The Frobenius covariant , for , is the matrix
It is essentially the Lagrange polynomial with matrix argument; alternatively, the same unique polynomial can be constructed using Newton polynomial. If the eigenvalue is simple, then as an idempotent projection matrix to a one-dimensional subspace, has a unit trace.
Computing the covariants
The Frobenius covariants of a matrix can be obtained from any eigendecomposition , where is non-singular and is diagonal with . The matrix is defined up to multiplication on the right by a diagonal matrix. If has no multiple eigenvalues, then let be the th right eigenvector of , that is, the th column of ; and let be the th left eigenvector of , namely the th row of −1. Then . As a projection matrix, the Frobenius covariant satisfies the relation
which leads to .
Given that and are the right and left vectors satisfying , the right and left eigenvectors of 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 .
If has an eigenvalue appearing multiple times, then , where the sum is over all rows and columns associated with the eigenvalue .
Example Consider the two-by-two matrix:
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 eigendecomposition is
Hence the Frobenius covariants, manifestly projections, are
with
Note , as required.
Generalization When a square matrix has repeated eigenvalues (such as a defective matrix or a diagonalizable one), the standard definition of Frobenius covariants based on Lagrange polynomials becomes undefined due to division by zero.
In this general case, let be the distinct eigenvalues of , each with an algebraic multiplicity of such that . The generalized Frobenius covariants are projection matrices ) that project onto the generalized eigenspace corresponding to along the direct sum of the remaining generalized eigenspaces, satisfying . Similarly, generalized Frobenius covariants enable the generalized Sylvester's formula to compute matrix functions for any square matrix.
The generalized Frobenius covariants can be constructed using several equivalent methods.
Construction via Hermite interpolation
To generalize the projection matrices for cases with repeated eigenvalues, one can use Hermite interpolating polynomials. The generalized Frobenius covariant is evaluated by applying the polynomial to as a matrix polynomial:
where is the degree- Hermite interpolating basis polynomial corresponding to node satisfying the interpolation conditions:
where denotes the Kronecker delta.
Let denote the th order derivative Hermite interpolating basis polynomial corresponding to node for . In terms of the Jordan–Chevalley decomposition, applying a higher-order basis polynomial to matrix isolates the th power of the nilpotent matrix associated with the Jordan blocks of (which is not a projection matrix for ). When , the th power of the nilpotent matrix acts as the identity on that generalized eigenspace, reducing to the projection matrix .
The degree- Hermite interpolating basis polynomial can be constructed using any standard Hermite interpolation method, such as divided differences with confluent nodes, solving a confluent Vandermonde system, or via the Chinese remainder theorem.
Closed-form explicit formula Based on the explicit formula for degree- Hermite interpolating basis polynomials, define the complementary factor polynomial for eigenvalue as . The degree- Hermite interpolating basis polynomial can be expressed directly in closed form:
where denotes the th degree Taylor operator of a function centered at . Evaluating this degree- Hermite interpolating basis polynomial at constructs the generalized Frobenius covariant .
Construction without Hermite interpolation Besides constructing Hermite interpolating basis polynomials, the generalized Frobenius covariants can also be constructed directly using several non-interpolation methods.
Jordan normal form This approach generalizes the eigendecomposition method for distinct eigenvalues described in to arbitrary matrices with non-trivial Jordan blocks. If the Jordan normal form decomposition is known, can be evaluated directly as , where is a block diagonal matrix whose blocks corresponding to are identity matrices of matching dimensions, with all other entries set to zero.
Nullspace projection Without finding Jordan chains, the generalized eigenspace of dimension and its complementary space of dimension decompose the vector space into a direct sum . In general, and are not orthogonal unless is a normal matrix.
Let be a matrix whose columns form a basis for , and be a matrix whose columns form a basis for . Assembling the partitioned basis matrix yields
Holomorphic matrix functional calculus Under the holomorphic matrix functional calculus, the generalized Frobenius covariant can be defined directly without polynomial interpolation as a contour integral of the resolvent matrix :
where is a simple closed contour in the complex plane that encloses in its interior while excluding all other eigenvalues of .
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