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What is the general procedure for doing so? For instance, how should one proceed after determining the roots and algebraic multiplicities? I also don't know when the superdiagonal is supposed to have nonzero entries. Any help on this matter would be appreciated.

chubbycantorset
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    There is a worked example here. Does it help? Once you have the eigenvalues (if the matrix is diagonalizable), you make up the matrices from a linear combination of the eigenvalues and eigenvectors. Here is a worked example (and there are many others) on MSE Regards – Amzoti Apr 12 '13 at 04:46

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Nonzero entries in the superdiagonal indicate the beginning of a new cycle (of generalized eigenvectors). But knowing the characteristic and minimal polynomial is not enough to determine the Jordan form of the matrix. The characteristic polynomial gives you the distinct eigenvalues and the dimensions of the generalized eigenspaces corresponding to those eigenvalues. The minimal polynomial gives you the maximal length of a cycle (for each eigenvalue). There can be matrices that have different Jordan canonical forms but have the same characteristic and minimal polynomials.

Basically, after you've found the eigenvalues, you set out to find cycles of generalized eigenvectors. I'd recommend reading this from Stephen Friedberg's Linear Algebra. However, for a quick survey, Amzoti's link to the Wikipedia article is a very good alternative.

Weltschmerz
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If you know (and can factor) the characteristic and minimal polynomials, and also have determined the geometric multiplicities of the eigenvalues (dimensions of the eigenspaces) then you've got information that in many cases will allow you to know the shape of the Jordan canonical form, though not always. Assuming there is only one eigenvalue $\lambda$ (determining generalised eigenspaces reduces to this case), if the characteristic polynomial is $(X-\lambda)^n$, the minimal polynomial $(X-\lambda)^m$ and if the eigenspace has dimension $d$, then there are $d$ blocks, the largest of which has size $m$ and the sum of sizes is $n$. This will be sufficient to determine the sizes of the blocks whenever $n\leq6$; the smallest case where it does not suffice is for $(d,m,n)=(3,3,7)$, where $(3,3,1)$ and $(3,2,2)$ are two possibilities for the sequence of Jordan block sizes.

To solve the remaining cases, you need to compute the dimension of the kernel of $(A-\lambda I)^i$ for all values $0<i<m$; this will tell you the number of block that have size at least $i$. Determining a basis on which the Jordan form is obtained is more difficult to describe in the general case, but it is hardly ever really necessary.

Marc van Leeuwen
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