Linear algebra done dubiously

A book that has been a contentious topic of discussion is Linear Algebra Done Right, by Axler. The reason, at least ostensibly[1], is because Axler’s treatment avoids the discussion of determinants. For the critics’ part, Axler himself seems to play this up, marketing the book as a revolutionary treatment where determinants are not discussed. Apparently, Sergei Treil found this marketing so offensive that he wrote a competing textbook known as Linear Algebra Done Wrong.

I do not quite buy the hype here. There’s a whole chapter on determinants in Axler’s book, which even includes a discussion of Jacobian determinants. Axler just doesn’t use determinants to prove the three main theorems of intermediate linear algebra over an algebraically closed field \overline K, namely the fact that every linear operator has an eigenvalue, that every linear operator has a unique Jordan canonical form, and the Cayley-Hamilton theorem. In all of these cases, one could prove the theorem using determinants, but there’s no good reason to, since there is a perfectly reasonable structure theory of linear operators over \overline K which does not mention determinants, and it gives fairly easy and conceptual proofs to all three theorems.

(I don’t think Axler’s book is perfect, for the record. Most annoyingly, he doesn’t seem to clearly distinguish between theorems that are valid over general \overline K and theorems that are specifically valid over \overline K = \mathbb C, which is the case for most of the results in the latter half of the book, except for one single chapter about the structure theory over \mathbb C. But I do think that a lot of the angry comments I’ve seen about the book on Reddit and elsewhere, which mainly focus on the issue of determinants, are just totally out to lunch.)

Anyways, it occurred to me today that the way I like to think about linear algebra neither involves determinants nor Axler’s structure theory, but is rather a complex-analytic version of linear algebra. I don’t think it essentially uses complex analysis though, and could probably be adopted to general \overline K.

The point is to consider the resolvent R(z) = (T - z)^{-1} of the linear operator T acting on a vector space V of dimension n, which is a rational map from \mathbb P^1 to a space of matrices. Clearly an eigenvalue is a pole of R, and the number of poles equals the number of zeroes (this is clearly true when \overline K = \mathbb C, but I suspect it is true for arbitrary \overline K). Since R has a zero of order n at \infty, T must have n eigenvalues. (If \overline K = \mathbb C, Rouche’s theorem even gives a bound on the size of the eigenvalues, and a way to compute approximations to the eigenvalues.)

Now we have n eigenvalues z_1, \dots, z_n counted with multiplicity. If \overline K = \mathbb C, we may consider loops \gamma_j around the puncture of \mathbb C \setminus \{z_j\} and define P_j = \frac{z_j}{2\pi i} \int_{\gamma_j} R(z) ~dz and similarly N_j = \frac{1}{2\pi i} \int_{\gamma_j} (z - z_j)R(z) ~dz. It is now a straightforward consequence of Cauchy’s integral formula that N_j is nilpotent and we have A = \sum_j P_j + N_j. Furthermore, if V_j is the image of P_j, then A acts on V_j as A = \lambda_j + N_j, and we have a direct sum decomposition V = \bigoplus_j V_j. That implies that A = \sum_j P_j + N_j is the Jordan canonical form of A. Let me leave the details to these notes of Knill. I would be very interested to see if an argument like this can be used in the general case of \overline K an algebraically closed field, possibly by replacing \gamma_j by the generator of some algebraic analogue of the fundamental group of the “open subscheme” (if that makes any sense) \overline K \setminus \{z_j\}, and replacing the differential form (z - z_j)R(z) ~\frac{dz}{2\pi i} with some sort of algebraic analogue of cohomology.

It remains to prove the Cayley-Hamilton theorem. (This proof, which was shown to me by Charles Pugh, is what got me thinking about linear algebra in this fashion in the first place.) Recall that the Cayley-Hamilton theorem says that if p is a characteristic polynomial of T, thus the zeroes of p are the eigenvalues of T, then p(T) = 0. This is obviously true if T is diagonalizable.

Now, the set of diagonalizable matrices is dense, because for example it includes the set of matrices with distinct eigenvalues, which is a generic set. On the other hand, the set Z of matrices with the Cayley-Hamilton property is closed, since p is continuous. Since clearly the space of all matrices is connected we conclude that Z = \overline K. This argument ostensibly works over \overline K = \mathbb C, but with a little work, it also holds for arbitrary \overline K, because we may use the Zariski topology.

This would be a pretty horrible way to teach linear algebra, but maybe one could simplify it so that it’s not so horrible.

[1] Axler has a signature, and quite clear and amicable, writing style, unlike most older textbooks. How much of the actual debate here is just Bourbakists in shambles?


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