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I met a linear algebra problem: Our field is complex number, find the Jordan basis for the matrix $$\begin{pmatrix} 0& 1 & 2&3\\ 0 &0&1&2\\ 0&0&0&1\\ 0&0&0&0\\ \end{pmatrix}$$

I think this problem is related Jordan Canonical Form, but I haven't learnt the term "Jordan basis". What is this? Thank you very much!

Another unrelated question: JCF and the rational form are useless in practice, right? SVD is more useful, right? I saw a lot of applications of SVD, like image compression, PCA, recommender systems, etc. But I have never, never, seen any applications of JCF or the rational form.

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  • $\begingroup$ what do you mean by "in practice"? both the JFC and the RCF are very useful when solving systems of differential/difference equations. see, for example, math.stackexchange.com/questions/655030/… , where I resort to the JCF to solve a system of linear difference equations. $\endgroup$ – etothepitimesi Mar 20 '15 at 3:24
  • $\begingroup$ You are right, it can be used to calculate matrix exponentials. Thank you. $\endgroup$ – breezeintopl Mar 20 '15 at 20:33
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A Jordan basis is a basis in which the matrix is in Jordan Canonical Form.

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  • $\begingroup$ Thank you for your answer! So, it is the collection of all the Jordan blocks(for each eigenvalue)? $\endgroup$ – breezeintopl Mar 20 '15 at 20:32

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