Usage examples of Jordan canonical form

Question

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?

Answer 1

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).

Answer 2

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.