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Continuo a non capire come mai ti ostini a scrivere le sommatorie con i puntini, invece che con il simbolo di sommatoria \(\sum\). Comunque, questo è un inizio per rendere un po' più pulito il codice.
\[\begin{align*}
(\mathbf{A}^{k}-\lambda^{k}\mathbf{I})\mathbf{x}_{j+1} &= \binom{k-1}{k-1}\lambda^{0}\mathbf{x}_{j-k+1} + \binom{k-1}{k-2}\lambda\mathbf{x}_{j-k+2} + \dots + \binom{k-1}{0}\lambda^{k-1}\mathbf{x}_{j}+\\
&+ \lambda \biggl[ \binom{k-2}{k-2}\lambda^{0}\mathbf{x}_{j-k+2} + \binom{k-2}{k-3}\lambda \mathbf{x}_{j-k+3} + \dots + \binom{k-2}{0}\lambda^{k-2}\mathbf{x}_{j} ]+\\
&+\lambda^{2} \biggl[\binom{k-3}{k-3}\lambda^{0}\mathbf{x}_{j-k+3} + \binom{k-3}{k-4}\lambda \mathbf{x}_{j-k+4}+ \binom{k-3}{0}\lambda^{k-3}\mathbf{x}_{j} \biggr]+\\
& \vdots \\
& +\lambda^{k-2} \biggl[\binom{1}{1} \lambda^{0}\mathbf{x}_{j-1} + \binom{1}{0}\lambda\mathbf{x}_{j}\biggr] + \lambda^{k-1}\binom{0}{0}\lambda^{0}\mathbf{x}_{j}\\
&= \binom{k-1}{k-1}\mathbf{x}_{j-k+1} + \lambda \mathbf{x}_{j-k+2} \biggl[ \binom{k-2}{k-2} + \binom{k-1}{k-2} \biggr]+\\ &+\lambda^{2}\mathbf{x}_{j-k+3} \biggl[\binom{k-3}{k-3} + \binom{k-2}{k-3} + \binom{k-1}{k-3}\biggr] + \\
& \vdots \\
& +\lambda^{k-2}\mathbf{x}_{j-1} \biggl[ \binom{1}{1} + \binom{2}{1}\dots + \binom{k-1}{1}\biggr]+\\
&+\lambda^{k-1}\mathbf{x}_{j} \biggl[ \binom{0}{0} + \binom{1}{0} + \dots + \binom{k-1}{0} \biggr]\\
& = \binom{k}{k}\mathbf{x}_{j-k+1} + \lambda \mathbf{x}_{j-k+2} \binom{k}{k-1} + \lambda^{2}\mathbf{x}_{j-k+3} \binom{k}{k-2} +\\
&\vdots\\
&+ \lambda^{k-2}\mathbf{x}_{j-1} \binom{k}{2} + \lambda^{k-1}\mathbf{x}_{j} \binom{k}{1}
\end{align*}\]

Codice:

\begin{align*}
 (\mathbf{A}^{k}-\lambda^{k}\mathbf{I})\mathbf{x}_{j+1}
 & = \binom{k-1}{k-1}\lambda^{0}\mathbf{x}_{j-k+1} + \binom{k-1}{k-2}\lambda\mathbf{x}_{j-k+2} + \dots + \binom{k-1}{0}\lambda^{k-1}\mathbf{x}_{j}+\\
 & + \lambda \biggl[ \binom{k-2}{k-2}\lambda^{0}\mathbf{x}_{j-k+2} +  \binom{k-2}{k-3}\lambda \mathbf{x}_{j-k+3} + \dots + \binom{k-2}{0}\lambda^{k-2}\mathbf{x}_{j} ]+\\
 & + \lambda^{2} \biggl[\binom{k-3}{k-3}\lambda^{0}\mathbf{x}_{j-k+3} + \binom{k-3}{k-4}\lambda \mathbf{x}_{j-k+4}+ \binom{k-3}{0}\lambda^{k-3}\mathbf{x}_{j} \biggr]+\\
 &   \vdots \\
 & + \lambda^{k-2} \biggl[\binom{1}{1} \lambda^{0}\mathbf{x}_{j-1} + \binom{1}{0}\lambda\mathbf{x}_{j}\biggr] + \lambda^{k-1}\binom{0}{0}\lambda^{0}\mathbf{x}_{j}\\
 & = \binom{k-1}{k-1}\mathbf{x}_{j-k+1} + \lambda \mathbf{x}_{j-k+2} \biggl[ \binom{k-2}{k-2} + \binom{k-1}{k-2} \biggr]+\\ &+\lambda^{2}\mathbf{x}_{j-k+3} \biggl[\binom{k-3}{k-3} + \binom{k-2}{k-3} + \binom{k-1}{k-3}\biggr] + \\
 &   \vdots \\
 &  +\lambda^{k-2}\mathbf{x}_{j-1} \biggl[ \binom{1}{1} + \binom{2}{1}\dots + \binom{k-1}{1}\biggr]+\\
 &  +\lambda^{k-1}\mathbf{x}_{j} \biggl[ \binom{0}{0} + \binom{1}{0} + \dots + \binom{k-1}{0} \biggr]\\
 &  =\binom{k}{k}\mathbf{x}_{j-k+1} + \lambda \mathbf{x}_{j-k+2}  \binom{k}{k-1} +  \lambda^{2}\mathbf{x}_{j-k+3} \binom{k}{k-2} +\\
 &   \vdots\\
 & + \lambda^{k-2}\mathbf{x}_{j-1}  \binom{k}{2} + \lambda^{k-1}\mathbf{x}_{j}  \binom{k}{1} 
\end{align*}

Continuo a non capire come mai ti ostini a scrivere le sommatorie con i puntini, invece che con il simbolo di sommatoria \(\sum\). Comunque, questo è un inizio per rendere un po' più pulito il codice.
\[\begin{align*}
(\mathbf{A}^{k}-\lambda^{k}\mathbf{I})\mathbf{x}_{j+1} &= \binom{k-1}{k-1}\lambda^{0}\mathbf{x}_{j-k+1} + \binom{k-1}{k-2}\lambda\mathbf{x}_{j-k+2} + \dots + \binom{k-1}{0}\lambda^{k-1}\mathbf{x}_{j}+\\
&+ \lambda \biggl[ \binom{k-2}{k-2}\lambda^{0}\mathbf{x}_{j-k+2} + \binom{k-2}{k-3}\lambda \mathbf{x}_{j-k+3} + \dots + \binom{k-2}{0}\lambda^{k-2}\mathbf{x}_{j} ]+\\
&+\lambda^{2} \biggl[\binom{k-3}{k-3}\lambda^{0}\mathbf{x}_{j-k+3} + \binom{k-3}{k-4}\lambda \mathbf{x}_{j-k+4}+ \binom{k-3}{0}\lambda^{k-3}\mathbf{x}_{j} \biggr]+\\
& \vdots \\
& +\lambda^{k-2} \biggl[\binom{1}{1} \lambda^{0}\mathbf{x}_{j-1} + \binom{1}{0}\lambda\mathbf{x}_{j}\biggr] + \lambda^{k-1}\binom{0}{0}\lambda^{0}\mathbf{x}_{j}\\
&= \binom{k-1}{k-1}\mathbf{x}_{j-k+1} + \lambda \mathbf{x}_{j-k+2} \biggl[ \binom{k-2}{k-2} + \binom{k-1}{k-2} \biggr]+\\ &+\lambda^{2}\mathbf{x}_{j-k+3} \biggl[\binom{k-3}{k-3} + \binom{k-2}{k-3} + \binom{k-1}{k-3}\biggr] + \\
& \vdots \\
& +\lambda^{k-2}\mathbf{x}_{j-1} \biggl[ \binom{1}{1} + \binom{2}{1}\dots + \binom{k-1}{1}\biggr]+\\
&+\lambda^{k-1}\mathbf{x}_{j} \biggl[ \binom{0}{0} + \binom{1}{0} + \dots + \binom{k-1}{0} \biggr]\\
& = \binom{k}{k}\mathbf{x}_{j-k+1} + \lambda \mathbf{x}_{j-k+2} \binom{k}{k-1} + \lambda^{2}\mathbf{x}_{j-k+3} \binom{k}{k-2} +\\
&\vdots\\
&+ \lambda^{k-2}\mathbf{x}_{j-1} \binom{k}{2} + \lambda^{k-1}\mathbf{x}_{j} \binom{k}{1}
\end{align*}\]

Codice:

\begin{align*}
 (\mathbf{A}^{k}-\lambda^{k}\mathbf{I})\mathbf{x}_{j+1}
 & = \binom{k-1}{k-1}\lambda^{0}\mathbf{x}_{j-k+1} + \binom{k-1}{k-2}\lambda\mathbf{x}_{j-k+2} + \dots + \binom{k-1}{0}\lambda^{k-1}\mathbf{x}_{j}+\\
 & + \lambda \biggl[ \binom{k-2}{k-2}\lambda^{0}\mathbf{x}_{j-k+2} +  \binom{k-2}{k-3}\lambda \mathbf{x}_{j-k+3} + \dots + \binom{k-2}{0}\lambda^{k-2}\mathbf{x}_{j} ]+\\
 & + \lambda^{2} \biggl[\binom{k-3}{k-3}\lambda^{0}\mathbf{x}_{j-k+3} + \binom{k-3}{k-4}\lambda \mathbf{x}_{j-k+4}+ \binom{k-3}{0}\lambda^{k-3}\mathbf{x}_{j} \biggr]+\\
 &   \vdots \\
 & + \lambda^{k-2} \biggl[\binom{1}{1} \lambda^{0}\mathbf{x}_{j-1} + \binom{1}{0}\lambda\mathbf{x}_{j}\biggr] + \lambda^{k-1}\binom{0}{0}\lambda^{0}\mathbf{x}_{j}\\
 & = \binom{k-1}{k-1}\mathbf{x}_{j-k+1} + \lambda \mathbf{x}_{j-k+2} \biggl[ \binom{k-2}{k-2} + \binom{k-1}{k-2} \biggr]+\\ &+\lambda^{2}\mathbf{x}_{j-k+3} \biggl[\binom{k-3}{k-3} + \binom{k-2}{k-3} + \binom{k-1}{k-3}\biggr] + \\
 &   \vdots \\
 &  +\lambda^{k-2}\mathbf{x}_{j-1} \biggl[ \binom{1}{1} + \binom{2}{1}\dots + \binom{k-1}{1}\biggr]+\\
 &  +\lambda^{k-1}\mathbf{x}_{j} \biggl[ \binom{0}{0} + \binom{1}{0} + \dots + \binom{k-1}{0} \biggr]\\
 &  =\binom{k}{k}\mathbf{x}_{j-k+1} + \lambda \mathbf{x}_{j-k+2}  \binom{k}{k-1} +  \lambda^{2}\mathbf{x}_{j-k+3} \binom{k}{k-2} +\\
 &   \vdots\\
 & + \lambda^{k-2}\mathbf{x}_{j-1}  \binom{k}{2} + \lambda^{k-1}\mathbf{x}_{j}  \binom{k}{1} 
\end{align*}

Ciao! Guarda ho provato ad utilizzare questo, ma nella pagina precedente a dove inserisco la formula mi lascia spazi enormi fra le varie cose, è sempre lo stesso problema che avevo all'inizio

\allowdisplaybreak nel preambolo

Grazie, provo subito