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I would like to know what the Jordan canonical form is useful for.

This answer says that “It simplifies many abstract proofs to assume a matrix in the proof is in Jordan canonical form”.

Can you give me a concrete example?

Martin Sleziak
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Hayatsu
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    Given a square matrix $A,$ if you can actually find a nonsingular matrix $P$ such that $P^{-1} A P = J$ is in Jordan form, then you can explicitly write $A^n$ for recurrences and $e^{At} $ for differential equation systems. As exercises, often $A$ and its eigenvalues are integers; it is worth practicing finding $P$ in such cases. – Will Jagy Nov 30 '22 at 02:26
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    Let $A$ be a complex $n\times n$-matrix. Then, $A^m \to 0$ as $m \to \infty$ if and only if all eigenvalues of $A$ have absolute value $< 1$. This famous fact (which is used in the theory of the spectral norm) is easiest to prove using the Jordan canonical form (see Corollary 3.5.2 in http://www.cip.ifi.lmu.de/~grinberg/t/21fala/alanotes.pdf for example; sorry for the mess that is Chapter 3 of these notes at the present). I did at some point figure out a Jordan-free proof, but it is in a way more complicated and certainly less memorable. – darij grinberg Nov 30 '22 at 02:40
  • When doing applications that involve matrices, like discrete time evolutions or continuous time evolutions, if the matrix is diagonalizable the problem becomes very easy to analyze, but if the matrix is not diagonlizable things are more difficult. But the Jordan canonical form gives a not-quite-as-easy-but-almost-as-easy way to analyze the problem in that situation (if the characteristic polynomial splits). – Arturo Magidin Nov 30 '22 at 14:54
  • Putting a matrix $A$ into Jordan canoncical form makes it relatively easy to compute $A^n$ for any integer $n$. – Mike Earnest Dec 05 '22 at 19:25
  • @darijgrinberg My shameless plug: I think my proof, based on the squeezing principle, is short and conceptually simpler (but surely, simplicity is a to-each-their-own thing). – user1551 Dec 09 '22 at 07:24

2 Answers2

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One example is that Jordan form affords a fairly quick proof of the (surprisingly non-obvious) fact that every matrix is similar to its transpose (link).

Erick Wong
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    Hmm, but this is usually (and arguably better) done using rational canonical form, because we don't need to consider the complications coming from field extension. – user1551 Dec 09 '22 at 07:46
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Two examples:

  • Jordan-Chevalley decompostion. Every square matrix $A$ over an algebraically closed field can be written as a sum of two commuting parts, one semi-simple and the other nilpotent. That is, we may write $A=D+N$ where $D$ is diagonalisable, $N$ is nilpotent and $DN=ND$. This is obvious if you look at the Jordan form $J$ of $A$: just take $D$ and $N$ as respectively the diagonal part and the strictly upper triangular part of $J$.
  • We all know that real symmetric matrices have very nice spectral properties and likewise for all Hermitian matrices. What about complex symmetric matrices? It turns out that there is nothing remarkable, because every complex square matrix is similar to some complex symmetric matrix. To prove this assertion, since every complex square matrix is similar to its Jordan form, it suffices to show that each Jordan block is similar to a complex symmetric matrix, but this is only a short exercise in Horn and Johnson's Matrix Analysis (see the first exercise on p.271 in the second edition and theorem 4.4.24 on the same page).

We may also use Jordan forms, not to simplify proofs, but to characterise some matrix properties. Two examples:

  • A complex matrix is similar to a real matrix iff the Jordan blocks for non-real eigenvalues in its Jordan form can be grouped into conjugate pairs.
  • A nonsingular real matrix $A$ has a real matrix square root or a real matrix logarithm iff in its Jordan normal form over $\mathbb C$, every Jordan block corresponding to a negative eigenvalue occurs an even number of times. See Walter J. Culver, On the existence and uniqueness of the real logarithm of a matrix, Proceedings of the American Mathematical Society, 17(5): 1146-1151, 1966.
user1551
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