In matrix theory, the Frobenius covariants of a square matrix A are special polynomials of it, namely projection matrices Ai 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 λi.
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.
^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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In matrix theory, the Frobeniuscovariants of a square matrix A are special polynomials of it, namely projection matrices Ai associated with the eigenvalues...
the Laurent series of f at a: It is proportional to the relevant Frobeniuscovariant. Then the sum St of the Qa,t, where a runs over all the roots of...
theorem Lebesgue constant The Chebfun system Table of Newtonian series Frobeniuscovariant Sylvester's formula Finite difference coefficient Hermite interpolation...
Hermitian definite pencil or definite pencil. Eigenvalue perturbation Frobeniuscovariant Householder transformation Jordan normal form List of matrices Matrix...
Sylvester's formula and the matrix-valued Lagrange polynomials are the Frobeniuscovariants. For a polynomial p n {\displaystyle p_{n}} of degree less than or...
which are required to be equivariant in the principal Lie group action. A covariant derivative in differential geometry is a linear differential operator...
{\displaystyle \to } S-Mod and G:S-Mod → {\displaystyle \to } R-Mod are additive (covariant) functors, then F and G are an equivalence if and only if there is a balanced...
value of any eigenvalue α of Frobenius on a fiber of E as follows. For any integer k, αk is an eigenvalue of Frobenius on a stalk of Ek, which for k...
_{r}=\int _{a}^{b}r(x)u(x)v(x)\,dx.} A double-dot product for matrices is the Frobenius inner product, which is analogous to the dot product on vectors. It is...
i\leq j} . A direct system over I {\displaystyle I} is then the same as a covariant functor I → C {\displaystyle {\mathcal {I}}\rightarrow {\mathcal {C}}}...
D {\displaystyle {\mathcal {D}}} is a pair of functors (assumed to be covariant) F : D → C {\displaystyle F:{\mathcal {D}}\rightarrow {\mathcal {C}}}...
connection. Suppose too we have a smooth section f of this bundle. Then the covariant derivative of f with respect to the connection is a smooth linear map...
− ) , {\displaystyle \mathrm {Hom} _{R}(-,-),} the tensor functor is covariant in both inputs. It can be shown that M ⊗ R − {\displaystyle M\otimes _{R}-}...
morphism is invertible. A (left) group action is then nothing but a (covariant) functor from G to the category of sets, and a group representation is...
the Lie bracket). In the language of category theory, we then have a covariant functor from the category of Lie groups to the category of Lie algebras...
fields with vanishing covariant derivative. In particular, the following facts hold: Hol(ω) ⊂ U(n) if and only if M admits a covariantly constant (or parallel)...
dyadics, we have: The first definition of the double-dot product is the Frobenius inner product, tr ( A B T ) = ∑ i , j tr ( a i b i T d j c j T ) =...
_{i}^{M}\|_{F},} where ‖ ⋅ ‖ F {\displaystyle \|\cdot \|_{F}} is the Frobenius norm. It was shown in a 2008 paper by de Silva and Lim that the above...
and we can always extend the results to the full Lie group through the Frobenius theorem. [ E , P i ] = 0 {\displaystyle [E,P_{i}]=0} [ P i , P j ] = 0...
direct sum of copies of the scalar ring R {\displaystyle R} . Frobenius reciprocity Frobenius reciprocity. Galois A Galois module is a module over the group...
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