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I don't understand why it's apparently 'clear' that the matrix of $T$ with respect to the basis $v_1, \dots, v_n$ is a Jordan block of degree $n$ if and only if $v_1, \dots ,v_n$ is a Jordan chain for $A$. Can anyone explain this? Thanks!

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What are your definitions? For instance, I have just learnt Jordan canonical form and some how never came across this term "Joran Chain". I might be able to help if I know what this means. Regards, –  user21436 Mar 21 '12 at 14:59
    
How do you define a Jordan chain? $(A-\lambda I)^n v_n=0$, and $(A-\lambda I)^k v_n=v_{n-k}$? –  Álvaro Lozano-Robledo Mar 21 '12 at 15:00
    
Ah I thought the notation was standard, sorry! A we define the Jordan chain as $(A - \lambda I)v_i = v_{i-1}$. –  user26069 Mar 21 '12 at 15:23

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Suppose that $F$ is a field, $V$ is an $n$-dimensional vector space, and $T:V\to V$ is a $F$-linear map. Suppose that $v_1,\ldots, v_n$ is a basis of $V$ such that $(T-\lambda I)v_k=v_{k-1}$ for $1\leq k\leq n$, where $v_0=0$ is the zero vector. Then: $$(T-\lambda I)^k v_n=v_{n-k},$$ for all $1\leq k \leq n$ and, in particular, $$(T-\lambda I)v_1 = (T-\lambda I)(T-\lambda I)^{n-1}v_n = (T-\lambda I)^n v_n=0.$$ Therefore, $T(v_1)=\lambda v_1$ and $v_1$ is an eigenvector with eigenvalue $\lambda$. Moreover, $(T-\lambda I)v_{k}=v_{k-1}$ and this implies $Tv_k = \lambda v_k + v_{k-1}$, for each $1\leq k\leq n$. Therefore, the matrix for $T$ in terms of the basis $\{v_1,\ldots,v_n\}$ of $V$ has the form: $$\left(\begin{array}[ccccccc] & \lambda & 1 & 0& 0 & \cdots & 0& 0 \\ 0 & \lambda & 1 & 0 & \cdots &0 & 0\\ 0 & 0 & \lambda & 1 & \cdots &0 & 0\\ \vdots & \vdots & \vdots & \vdots &\ddots & \vdots & \vdots\\ 0 & 0 & 0 & 0 & \cdots &1 & 0\\ 0 & 0 & 0 & 0 & \cdots &\lambda & 1\\ 0 & 0 & 0 & 0 & \cdots & 0 & \lambda\\ \end{array}\right),$$ which is clearly a Jordan block.

Conversely, if the matrix of $T$ is a Jordan block as above with respect to a basis $\{v_1,\ldots, v_n\}$, then $T(v_1)=\lambda v_1$ and $T(v_k)=\lambda v_k + v_{k-1}$, or equivalently, $(T-\lambda I)v_1=0$ and $(T-\lambda I)v_k=v_{k-1}$, so the vectors in the basis form a Jordan chain.

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Writing $T(v_i)$ as a linear combination gives you the column of the matrix of $T$ with respect to this Basis. But for a Jordan chain $$ T(v_1)=\lambda v_1 \quad T(v_2)=v_1 + \lambda v_2 \quad T(v_3)=v_2+\lambda v_3 \dots $$

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