A Note on the Jordan Canonical Form

A proof of the Jordan canonical form, suitable for a first course in linear algebra, is given. The proof includes the uniqueness of the number and sizes of the Jordan blocks. The value of the customary procedure for finding the block generators is also questioned. 2000 MSC: 15A21. The Jordan form of linear transformations is an exceeding useful result in all theoretical considerations regarding conjugacy classes of matrices, nilpotent orbits and the Jacobson -Morozov theorem. The author wishes to share a proof of the Jordan form which he found in connection with a problem in Lie theory. The ideas of the proof give at the same time the number and sizes of all the blocks. The proof has the added advantage that the most important parts can be taught in a first course on linear algebra, as soon as basic ideas have been introduced and the invariance of dimensions has been established. It is thus also a contribution to the teaching of these ideas. Although extensive work has been done in [4] regarding this circle of ideas, our method provides a very simple algorithm whose importance is shown through some simple examples. A classical reference for this topic is Smirnov’s book [2,p.245-254]. There is a very well known proof due to Fillipov [1], which is also given in Strang’s book [3, p. 422425]. There is also a proof given in the Wikipedia [5]. The proofs in [3 & 5] do not give sufficient details regarding the number of Jordan blocks and their sizes, nor an algorithmic procedure to handle matrices of large size. In view of the algorithm 1 given in this note, and the examples given below, it is not clear to us why a precise determination of the block generators is needed, although, for the sake of completeness, we have discussed this aspect tooat the expense of increasing the level of exposition. It would be very desirable to compare the computational complexity of computing the Jordan canonical form, using the algorithm given in this paper, with other algorithms, which programmes like Maple and Mathematica use to determine the Jordan form. As is well known, the main technical step in establishing the Jordan canonical form is to prove its existence and uniqueness for nilpotent transformations. We will return to the general case towards the end of this note. Let A be a nilpotent transformation on a finite dimensional vector space V , let v be a nonzero vector in V and n the smallest integer such that Av = 0. Proposition 1 The vectors {Aiv : 0 ≤ i < n} are linearly independent. Proof. Take an expression n−1 ∑