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Upper triangular matrix/Finding Jordan normal form/Generalized eigenspace/Method

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We describe how to find to a linear trigonalizable mapping a basis, such that describing matrix with respect to this basis is in Jordan normal form. For this, we determine, for every eigenvalue , the minimal exponent with

This kernel is the generalized eigenspace to . We set

for . This yields the chain

Now, we choose a vector from . The vectors

form a basis for a Jordan block. If this basis generates the generalized eigenspace, then we are done. Otherwise, we look in for another vector which is linearly independent to and to . Again, we add this vector and all its successive images. If is exhausted, then we look whether is already covered, and so on. If the generalized eigenspace to is covered, then we continue with the next eigenvalue.

Under certain circumstances, we can also start with a basis of the eigenspace. If, for example, the eigenspace to is one-dimensional, then we can choose an eigenvector for , and we can find successively preimages under of the vectors, that is, we have to solve the equation

then

etc.

If, for example, the eigenspace is -dimensional and the generalized eigenspace is -dimensional, then we only have to find a preimage for one eigenvector under .