Matrices $ B in mathbf {C} ^ {n times n} $ move with a given matrix $ A in mathbf {C} ^ {n times n} $ in Jordan it is known that the normal form is constructed from Toeplitz's superior triangular matrices. I have seen convincing evidence, but I wanted to derive this fact by my own method (in Dirac notation):
 Consider the base of generalized eigenbasis (Jordan base) for A and its adjunct $ A ^ daga $ belonging to selfworth $ lambda $:
$  R_i rangle in ker (A lambda) ^ i ker (A lambda) ^ {i1},  L_j rangle in ker (A ^ dagger lambda ^ *) ^ i ker (A ^ dagger lambda ^ *) ^ {i1} $
giving rise to the Jordanian chains.

Squeeze $ B $ in this base (right and left): $ B = sum_ {ij} B_ {ij}  R_i rangle langle L_j  $

Apply $ A $ on the left resp. Right:
$ AB = sum_ {ij} B_ {ij} A  R_i rangle langle L_j  = sum_ {ij} B_ {ij} lambda  R_i rangle langle L_j  + sum_ {ij} B_ {ij}  R_ {i1} rangle langle L_j  $
$ BA = sum_ {ij} B_ {ij}  R_i rangle langle L_j  A = sum_ {ij} B_ {ij} lambda  R_i rangle langle L_j  + sum_ {ij} B_ {ij}  R_i rangle langle L_ {j1}  $
 As $  R_i rangle langle L_j  $ It is a basis for the space of matrices, equality $ AB = BA $ is equivalent to the equality of the respective coefficients separately, in particular:
$ sum_ {ij} B_ {ij}  R_ {i1} rangle langle L_j  = sum_ {ij} B_ {ij}  R_i rangle langle L_ {j1}  $
 Redefining the summation rates via $ k: = i1 $ Y $ l: = j1 $ yields
$ sum_ {kj} B_ {k + 1, j}  R_k rangle langle L_j  = sum_ {il} B_ {i, l + 1}  R_i rangle langle L_l  $ and therefore $ B_ {i + 1, j} = B_ {i, j + 1} $
But the Toeplitz matrices are characterized by $ B_ {i + 1, j} = B_ {i, j1} $! What's wrong here?