page_title: Frobenius covariant
section_title: Computing the covariants
hierarchical_section_title: Frobenius covariant / Computing the covariants
caption_attribution_description: Deutsch: Fotografie des Mathematikers Georg Frobenius English: Photography of the mathematician Georg Frobenius
context_page_description: 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. 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 as a matrix polynomial, namely a linear combination
  of that function's values on the eigenvalues of A.
context_section_description: 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 Dᵢ,ᵢ = λᵢ. 
  If A has no multiple eigenvalues, then let cᵢ be the ith right eigenvector of A, that is, the ith column of S; and let rᵢ be the ith left eigenvector of A, namely the ith row of S⁻¹.  Then Aᵢ = cᵢ rᵢ.
  If A has an eigenvalue λᵢ appear multiple times, then Aᵢ = Σⱼ cⱼ rⱼ, where the sum is over all rows and columns associated with the eigenvalue λᵢ.
caption: Ferdinand Georg Frobenius (1849–1917), German mathematician. His main interests were elliptic functions differential equations, and later group theory.
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