Recently there was some discussion in the n-category cafe about linear algebra and specially about determinants and their place in a linear algebra undergraduate course. The whole discussion was in fact triggered by a rather polemic paper entitled `Down with Determinants' which seemed to suggest that determinants better be introduced in the graduate school first! This `extremist' point of view was, understandably, challenged by several people who pointed out various algebraic and geometric aspects of determinants and their importance. You can also find a rebuttal in Lieven Le Bruyn's blog here. I would like to echo those sentiments and say: long live determinants!
I don't want to say much about determinants or how linear algebra should be taught to undergraduates. This is not what this blog is all about and in fact I am not sure I am qualified enough in that regard. Others are of course most welcome to comment on all aspects of these issues as they see fit. I just wanted to mention one aspect of determinants and its relation with traces that have some implications for NCG. But first a bit of early history of determinants is perhaps in order.
Determinants have a long history going back to some work of Leibniz and the Japanese mathematician Seki Kowa, also know as Takakazu, in the 17 th century. Cramer's rule of mid 18th century seems to be the first general result on determinants. In the 19th century Sylvester suggested determinants be called `Bezoutians' in honor of Etienne Bezout (see this newly published English translation of Bezout's old text, General Theory of Algebraic Equations ). For more history and in particular to get a glimpse of what happened in the 19th century see this Wikepedia article which also cites a 3rd century BC Chinese text!
Now the point of introducing determinants was not to show that matrices (over algebraically closed fields) have eigenvalues-this came later of course. There are equally interesting applications of determinants, however (see below for just a few), and postponing a proper introduction of determinants to graduate years I don't think would be wise.
As for the existence of eigenvalues for matrices or more generally the non-emptiness of the spectrum of an element of an algebra I know at least two very general situations where one can show that the spectrum of any element is non-empty. They both work in infinite dimensional situations and neither use a determinant function (which may not exist after all). When the algebra is a complex unital Banach algebra this result is due to Gelfand. In fact the whole notion of a Banach algebra is due to Gelfand from the late 1930's who called it normed rings. This was then used in a crucial way, by Gelfand and Naimark, in the proof of their celebrated theorem on the structure of commutative C*-algebras. A second case is an algebra over an algebraically closed field where the dimension of the algebra is less than the cardinality of the field. The argument in this case is similar to the proof of the existence of eigenvalues used in the `dwd' paper mentined above. Notice that the dimension of the algebra need not be finite now. This added generality is not a luxury and comes quite handy in proving things like Hilbert's Nullstellensatz (for fields with uncountable number of elements-see the first chapter of this book). Notice that Nullstellensatz is an algebraic analogue of the Gelfand-Naimark theorem and both results are pivotal for the general philosophy of noncommutative geometry.
On the educational side, see here for a nice story on `how to compute determinants', or perhaps how not to compute a determinant! Halmos' old (1940's?) book on finite dimensional vector spaces has gone through many editions but is still highly readable and one of my favorites. It is written with a view towards functional analysis and operator theory on Hilbert space. So one learns early enough about spectral theorem, polar decomposition, and determinants which are defined using the exterior algebra and volume forms. It does not cover much in multilinear algebra though. Another favorite of mine is Manin and Kostrikin's more modern and wonderful book Linear Algebra and Geometry . It covers a lot of topics well beyond the standard stuff on canonical forms. Things such as multilinear algebra, determinants and Pfaffians, reverse triangle inequality in Minkowski space and the twin paradox, foundations of quantum mechanics, and fast multiplication algorithms. It has even some small item on Feynman rules in QFT! Another favourite of mine is Prasolov's problems and theorems in linear algebra. You should also definitely check P. Cartier's A course on determinants (in: Conformal Invariance and String Theory, 1987) for a nice and modern survey of determinants. It covers much including things like infinite dimensional Fredholm determinants, and superdeterminants.
In the 19th century when people wanted to prove that the sum and product of algebraic numbers is again an algebraic number they would use determinants. Nowadays of course this is done using vector spaces and the formula dim_E K=(dim_E F)( dim_F K) for field extensions E C F C K. But imagine you really want to find a polynomial with rational coefficients P such that P (a+b)=0, or P(ab)=0, assuming a and b are algebraic. What would you do? Here is a modern adaptation of the classical method to find such a polynomial P quickly. Notice that a complex number is algebraic iff it is the eigenvalue of a matrix with rational coefficients. If a and b are eigenvalues of A and B, then a+b is an eigenvalue of I \otimes B + A\otimes I (remember the Hamiltonian of a combined system of two particles in QM?) and ab is of course an eigenvalue of A \otimes B. We should then just compute the characteristic polynomials of these matrices which is pretty straightforward. A similar proof applies to show that if a and b are algebraic integers then so are ab and a+b.....
Determinants and Physics: Pauli's exclusion principle in quantum mechanics can be formulated mathematically as saying that if the Hilbert space of states of a fermion is H then the Hilbert space of states of a pair of fermions should be H /\H, the exterior product of H with itself. By the same principle the Hilbert space of n fermions should be the n-th exterior power of H, /\^n H. For bosons, on the other hand, the appropriate n particle Hilbert space is the n-th symmetric power S^n H. Now, mandated by the special theory of relativity, quantum field theory and the second quantization of fields tell us that the number of particles can not remain constant and so, in the case of fermions, we go over to the so called fermionic Fock space
/\ H = C+ H + H/\H + H/\H/\H +..........
An operator A: H -> H induces an operator
/\ A : /\ H -> /\ H, (/\ A) (v_1 /\v2 ....../\v_n)= Av_1 /\Av_2 ..../\Av_n
Similarly for bosons the apprpriate Hilbert space is the bosonic Fock space
SH =C+ H + S^2 H + S^3 H +......
with the associated operator
SA: SH -> SH
Now if H is n-dimensional, /\^n V is one dimnsional and we see that the exterior algebra gives us a formula/definition for the determinant in terms of trace:
Det (A) = Tr (/\^n A)
We see a direct link here between the exterior algebra, fermions, and determinants. The above formula is the beginning of a series of formulas relating the determinant and trace. For example the beautiful MacMahon Master Theorem states that
(1) Det (1+tA) =\sum t^k Tr (/\^k A)
and if we put t=-1 we obtain
(2) Det (1-A) =\sum (-1)^k Tr (/\^k A)
To prove this one can first assume A is diagonal(izable) in which case the proof is really easy and then use the fact that diagonalizable matrices are dense among all matrices, plus continuity and invariance under conjugation of both sides. There are of course more algebraic proofs that work over any commutative ring.
We would like to write the RHS of (2) as trace of /\A acting on /\ V but because of signs it can not be the usual trace. Instead we can invoke the fact that the fermionic Fock space /\V is a super vector space graded by degree of tensors. We can define the Supertrace of an operator A as Tr_s (A)= Tr (A^+) - Tr (A^-) with A^+ and A^- designating the even and odd parts of A and with this understood we can write (2) as
(3) Det (1-A) =Tr_s (/\ A)
There is a similar formula for the bosonic second quantization and this time we have
(4) [Det (1-tA)]^-1 =\sum t^k Tr (S^k A)
If we put t=1 we obtain the beautiful formula
(5) [Det (1-A)]^-1 = Tr (SA)
Combining formulas (3) and (5) we obtain the boson-fermion formula
(6) Tr_s(/\ A) Tr (SA)=1
which puts in duality the exterior algebra with symmetric algebra. I have completely bypassed the convergence issues which are relevant even when H is finite dimensional since the bosonic Fock space SH is infinte-dimensional but this can be managed relatively easily....
For a particular choice of H and A, Bost and Connes in their paper `Hecke algebras, type III factors and phase transitions with spontaneous symmetry breaking in number theory. Selecta Math. (N.S.) 1 (1995), no. 3, 411--457, right in the begining show that the above formula (5) gives the Euler product formula for the zeta function (see also page 529 of this book) In fact their paper starts by quantizing the set of prime numbers by finding a natural operator whose bosonic second quantization has the set of primes as its spectrum (I leave to you as an exercise the task of finding this operator) and this is the begining of their long journey towards understanding the statistical behaviour of primes using tools of quantum mechanics and noncommutative geometry.
Another interesting issue with regard to the boson-fermion duality formula (6) is its relation with Koszul duality. The Koszul dual of the quadratic algebra /\ H is the symmetric algebra S H. I suppose it is possible to give a proof of (6) starting from Koszul duality but this will need another post and I certainly hope others who know more will jump in and enlighten us. The same with q-analogues of (6). There are many other interesting things that remain to be said things like Pfaffians as fermionic Gaussian integrals and their supersymmetric and Clifford algebra analogues, regularized determinants, etc. Hopefully others will comment on aspects of determinants and their applications in their work.
Enter Trace Unlike the determinant, trace has a nice straightforward extension to the noncommutative world. In fact trace is the real queen of noncommutative geometry! For any algebra A, commutative or not, we have a trace map
Tr: M_n(A) -> T(A)
From the algebra of n by n matrices with entries in A to the quotient of A by the linear span of commutators in A. It is, be definition, the sum of the elements on the main diagonal followed by the quotient map. It has the trace property in the sense that it is linear (over C) and satisfies T(xy)=T(yx) for all x and y. It is also easy to see that Tr is indeed the universal trace on A in the sense that any other trace tr: A ->V factors through it. This can be extended a bit. Let E be a finitely generated projective right A-module and let End_A (E) denote the algebra of A-linear maps from E to E. Then there is a trace map
Tr: End_A (E) -> T(A).
It can be defined by first embedding E into a finite and free A-module and then using the above trace. Alternatively one can use the fact that End_A (E) =E\otimes_A E* and then use the standard dual pairing between E and its dual E* to land in A and then apply the quotient map.
This allows us to extend the notion of dimension from commutative to noncommutative geometry. The word dimension is loaded with many meanings and interpretations and here we just look at one of those. Let us fix a C-valued trace tr on A. The classical formula
Dim (E) = tr (id_E)
relates the dimension of a vector space or the fiber dimension of a vector bundle to trace and is integer valued in that context. When we use this formula as the definition of the dimension of a finite projective module (aka noncommutative vector bundles) we should be prepared to see non-integral dimensions! (See page 361 of Alain's 1994 book for an example).
Remark: Apart from Hausdorff dimension, this sort of continuous dimensions were first investigated by von Neumann in a purely algebraic and synthetic manner in his book continuous geometry but then his dream of a continuous geometry was, partially, realized in his theory of von Neumann algebras. We say partially because it covered only the measure theoretic aspects of the noncommutative world. The full dream was only realized by the advent of NCG!