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So I am trying to compile a summary of the procedure one should follow to find the Jordan basis and the Jordan form of a matrix, and I am on the lookout for free resources online where the algorithm to be followed is clearly explained in an amenable way.

I have found some interesting youtube videos but what I am on the lookout for is a written thing.

Alternatively, if you are so kind as to flesh out the procedure I would be happy to accept that as an answer.

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  • Find all the eigenvalues of $T$.

  • For each eigenvalue $\lambda$:

    • Let $N = T-\lambda I$.

    • Compute $N^2, N^3, \dots, N^n$.

    • Find the generalized eigenspace $G=G(\lambda,T)$ of solutions $u$ to $N^n u = 0$.

    • Find a temporary basis for $G$.

    • Let $U_0 = G$, $U_n = \{0\}$ and $B_n = \emptyset$.

    • For $k=n-1,\dots,1,0$:

      • Find $U_k = $range$ (N_{|_G})^k$ by applying $N^k$ to the temporary basis of $G$.

      • From the previous step we have a Jordan basis $B_{k+1}$ to $T_{|_{U_{k+1}}}$ given by $N^{d_1}v_1,\dots,N^2 v_1, N v_1, v_1, \dots, N^{d_m}v_m,\dots,N^2 v_m, N v_m, v_m$, with the property that $N^{d_j+1} v_j = 0$ for all $j$.

      • For $j=1,\dots,m$, find one $u_j$ such that $N u_j = v_j$.

      • Let $\tilde B_k = N^{d_1}v_1,\dots,N^2 v_1, N v_1, v_1, u_1 \dots, N^{d_m}v_m,\dots,N^2 v_m, N v_m, v_m, u_m$. Then $\tilde B_k$ is a Jordan basis for $T$ restricted to its span.

      • Find $A_k$ such that $\tilde B_k \cup A_k$ is a basis for $U_k$.

      • For each $w\in A_k$:

        • Find $x \in {\sf span} \tilde B_k$ such that $Nx = Nw$.

        • Let $u=w-x$, so $Nu=0$.

      • Let $\tilde A_k$ be the set of vectors obtained above.

      • Let $B_k = \tilde B_k \cup A_k$. Then $B_k$ is a Jordan basis for $T_{|_{U_k}}$.

    • In the end, $B_0$ is a Jordan basis for $T_{|_G}$.

  • Recollecting all Jordan bases for each $T_{|_{G(\lambda,T)}}$ produces a Jordan basis for $T$.

I found this method myself by digging into the proof in Axler's Done Right book. I hope it is correct.

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