In matrix theory, the Frobenius covariants of a square matrix A are matrices A_i associated with the eigenvalues and eigenvectors

Eigenvalue, eigenvector and eigenspace

The eigenvectors of a square matrix are the non-zero vectors that, after being multiplied by the matrix, remain parallel to the original vector. For each eigenvector, the corresponding eigenvalue is the factor by which the eigenvector is scaled when multiplied by the matrix...

 of A. Each covariant is a projection

Projection (linear algebra)

In linear algebra and functional analysis, a projection is a linear transformation P from a vector space to itself such that P2 = P. It leaves its image unchanged....

 on the eigenspace

Eigenvalue, eigenvector and eigenspace

The eigenvectors of a square matrix are the non-zero vectors that, after being multiplied by the matrix, remain parallel to the original vector. For each eigenvector, the corresponding eigenvalue is the factor by which the eigenvector is scaled when multiplied by the matrix...

 associated with λ_i.

Frobenius covariants are the coefficients of Sylvester's formula

Sylvester's formula

In matrix theory, Sylvester's formula or Sylvester's matrix theorem expresses an analytic function f of a matrix A in terms of the eigenvalues and eigenvectors of A...

, that expresses a function of a matrix

Matrix function

In mathematics, a matrix function is a function which maps a matrix to another matrix.- Extending scalar functions to matrix functions :There are several techniques for lifting a real function to a square matrix function such that interesting properties are maintained...

 f(A) as a linear combination of its values on the eigenvalues of A. They are named after the mathematician Ferdinand Frobenius

Ferdinand Georg Frobenius

Ferdinand Georg Frobenius was a German mathematician, best known for his contributions to the theory of differential equations and to group theory...

.

Formal definition

Let A be a diagonalizable matrix

Diagonalizable matrix

In linear algebra, a square matrix A is called diagonalizable if it is similar to a diagonal matrix, i.e., if there exists an invertible matrix P such that P −1AP is a diagonal matrix...

 with k distinct eigenvalues, λ₁, …, λ_k. The Frobenius covariant A_i, for i = 1,…, k, is the matrix

Computing the covariants

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 ith left eigenvector of A, that is, the ith column of S; and let r_i be the ith right eigenvector of A, namely the ith row of S⁻¹. Then A_i = c_ir_i.

If A has multiple eigenvalues then A_i = Σ_j c_jr_j, where the sum is over all rows and columns associated with the eigenvalue λ_i.

Example

Consider the two-by-two matrix:
This matrix has two eigenvalues, 5 and −2. The corresponding eigen decomposition is
Hence the Frobenius covariants are