Linear algebra – Jordan Matrix switch

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):

  1. Consider the base of generalized eigenbasis (Jordan base) for A and its adjunct $ A ^ daga $ belonging to self-worth $ lambda $:

$ | R_i rangle in ker (A- lambda) ^ i- ker (A- lambda) ^ {i-1}, | L_j rangle in ker (A ^ dagger- lambda ^ *) ^ i- ker (A ^ dagger- lambda ^ *) ^ {i-1} $

giving rise to the Jordanian chains.

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

  2. 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_ {i-1} 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_ {j-1} | $

  1. 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_ {i-1} rangle langle L_j | = sum_ {ij} B_ {ij} | R_i rangle langle L_ {j-1} | $

  1. Redefining the summation rates via $ k: = i-1 $ Y $ l: = j-1 $ 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, j-1} $! What's wrong here?