Frobenius covariant

In matrix theory, the Frobenius covariants of a square matrix are special polynomials of it, namely projection matrices "A""i" associated with the eigenvalues and eigenvectors of . 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 "λ"1, …, "λ""k".
The Frobenius covariant , for "i" = 1,…, "k", is the matrix
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, 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 . 
If has no multiple eigenvalues, then let "c""i" be the th right eigenvector of , that is, the th column of ; and let "r""i" be the th left eigenvector of , namely the th row of −1. Then .
If has an eigenvalue "λ""i" appear multiple times, then , 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; hence .
The corresponding eigen decomposition is
Hence the Frobenius covariants, manifestly projections, are 
with
Note , as required.