In the Jordan form of square matrix $A \longrightarrow T^{-1}AT = J$, $J$ needs to be upper bidiagonal; but should the upper diagonal be restricted to ones?.
The equations $Av_i = v_{i-1} + \lambda_iv_i $, where $v_i$ are the columns of $T$, result from the Jordan form and they establish the linear independence of $T$'s columns. Why cant we have $Av_i = 2v_{i-1} + \lambda_iv_i $ with upper diagonal being 2 or just any number.