# what is Jordan basis for a matrix? and is JCF useful in practice?

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.

• 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. – etothepitimesi Mar 20 '15 at 3:24
• You are right, it can be used to calculate matrix exponentials. Thank you. – breezeintopl Mar 20 '15 at 20:33