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Jordan Form

This is happening to thousands of Math graduate students all over the country:

You are taking a Linear Algebra class, and you are expected to be able to find the Jordan form of a matrix on the exams (and probably the qualifier). Your textbook (perhaps “Matrix Analysis” by Horn and Johnson?) describes the form, proves that every complex matrix is similar to a matrix in Jordan form, describes what can be extrapolated about a matrix given its Jordan form, all with detailed proofs. Then it goes on to hand wave about how to find the Jordan form of a matrix, and extrapolating the algorithm from the description takes a degree in re-constructive surgery. Your algebra textbook (Dummit & Foote?) has an even more confusing description.

The usual suspects like Wikipedia and Mathworld are completely useless. Google has never failed you like this before. Your textbook also explains why the Jordan form is completely useless in practical applications, which explains why nobody bothers to explain the algorithm and then requires you to know it.

You ask a senior student or your professor, and they explain about generalized eigenvectors. You grind through some examples, and finally it starts to make sense.

Or you stumble across this everything2 article, which is the only published article on or off the Internet describing the algorithm in plain language. Actually it’s still pretty confusing, but as a grad student, you have a high complexity (pain) tolerance.

It helps to realize that the process is similar to diagonalization (and in fact is diagonalization if the matrix is diagonalizable) but instead of placing the eigenvectors as columns in the transformation matrix, you place the generalized eigenvector as columns (and order matters to some extent here).

If you don’t need the transformation matrix, you may be able to skip some of these steps: the geometric multiplicity of an eigenvalue gives the number of Jordan blocks associated with that eigenvalue, the algebraic multiplicity gives the sum of the sizes of all Jordan blocks associated with an eigenvalue, and the size of the blocks can be found by looking at… er… how many generalized eigenvalues are associated with each eigenvector. I don’t have my notes here so I’m getting a bit hazy on that part. Perhaps I will make this clear in a later edit…

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