neverendingbooks

Here pdf-files of older NeverEndingBooks-posts on geometry. For more recent posts go here.

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The categorical cafe has a guest post by Tom Leinster Linear Algebra Done Right on the book with the same title by Sheldon Axler. I haven’t read the book but glanced through his online paper Down with determinants!. Here is ‘his’ proof of the fact that any n by n matrix A has at least one eigenvector. Take a vector $v \in \mathbb{C}^n $, then as the collection of vectors ${ v,A.v,A^2.v,\ldots,A^n.v } $ must be linearly dependent, there are complex numbers $a_i \in \mathbb{C} $ such that $~(a_0 + a_1 A + a_2 A^2 + \ldots + a_n A^n).v = \vec{0} \in \mathbb{C}^n $ But then as $\mathbb{C} $ is algebraically closed the polynomial on the left factors into linear factors $a_0 + a_1 x + a_2 x^2 + \ldots + a_n x^n = c (x-r_1)(x-r_2) \ldots (x-r_n) $ and therefore as $c(A-r_1I_n)(A-r_2I_n) \ldots (A-r_nI_n).v = \vec{0} $ from which it follows that at least one of the linear transformations $A-r_j I_n $ has a non-trivial kernel, whence A has an eigenvector with eigenvalue $r_j $. Okay, fine, nice even, but does this simple minded observation warrant the extreme conclusion of his paper (on page 18) ?

As mathematicians, we often read a nice new proof of a known theorem, enjoy the different approach, but continue to derive our internal understanding from the method we originally learned. This paper aims to change drastically the way mathematicians think about and teach crucial aspects of linear algebra.

The simple proof of the existence of eigenvalues given in Theorem 2.1 should be the one imprinted in our minds, written on our blackboards, and published in our textbooks. Generalized eigenvectors should become a central tool for the understanding of linear operators. As we have seen, their use leads to natural definitions of multiplicity and the characteristic polynomial. Every mathematician and every linear algebra student should at least remember that the generalized eigenvectors of an operator always span the domain (Proposition 3.4)—this crucial result leads to easy proofs of upper-triangular form (Theorem 6.2) and the Spectral Theorem (Theorems 7.5 and 8.3).

Determinants appear in many proofs not discussed here. If you scrutinize such proofs, you’ll often discover better alternatives without determinants. Down with Determinants!

I welcome all new proofs of known results as they allow instructors to choose the one best suited to their students (and preferable giving more than one proof showing that there is no such thing as ‘the best way’ to prove a mathematical result). What worries me is Axler’s attitude shared by extremists and dogmatics world-wide : they are so blinded by their own right that they impoverish their own lifes (and if they had their way, also that of others) by not willing to consider other alternatives. A few other comments :

1. I would be far more impressed if he had given a short argument for the one line he skates over in his proof, that of $\mathbb{C} $ being algebraically closed. Does anyone give a proof of this fact anymore or is this one of the few facts we expect first year students to accept on faith?

   1. I dont understand this aversity to the determinant (probably because of its nonlinear character) but at the same time not having any problems with successive powers of matrices. Surely he knows that the determinant is a fixed $~\mathbb{Q}~ $-polynomial in the traces (which are linear!) of powers of the matrix.

   2. The essense of linear algebra is that by choosing a basis cleverly one can express a linear operator in a extremely nice matrix form (a canonical form) so that all computations become much more easy. This crucial idea of considering different bases and their basechange seems to be missing from Axler’s approach. Moreover, I would have thought that everyone would know these days that ‘linear algebra done right’ is a well developed topic called ‘representation theory of quivers’ but I realize this might be viewed as a dogmatic statement. Fortunately someone else is giving the basic linear algebra courses here in Antwerp so students are spared my private obsessions (at least the first few years…). In [his post](http://golem.ph.utexas.edu/category/2007/05/ linear_algebra_done_right.html) Leistner askes “What are determinants good for?” I cannot resist mentioning a trivial observation I made last week when thinking once again about THE rationality problem and which may be well known to others. Recall from the previous post that rationality of the quotient variety of matrix-couples $~(A,B) \in M_n(\mathbb{C}) \oplus M_n(\mathbb{C}) / GL_n $ under _simultaneous conjugation_ is a very hard problem. On the other hand, the ‘near miss’ problem of the quotient variety of matrix-couples $ { (A,B)~|~det(A)=0~} / GL_n $ is completely trivial. It is rational for all n. Here is a one-line proof. Consider the quiver $\xymatrix{\vtx{} \ar@/^2ex/[rr] & & \vtx{} \ar@(ur,dr) \ar@/^2ex/[ll]} $ then the dimension vector (n-1,n) is a Schur root and the first fundamental theorem of $GL_n $ (see for example Hanspeter Krafts excellent book on invariant theory) asserts that the corresponding quotient variety is the one above. The result then follows from Aidan Schofield’s paper Birational classification of moduli spaces of representations of quivers. Btw. in this special case one does not have to use the full force of Aidan’s result. Zinovy Reichstein, who keeps me updated on events in Atlanta, emailed the following elegant short proof Here is an outline of a geometric proof. Let $X = {(A, B) : det(A) = 0} \subset M_n^2 $ and $Y = \mathbb{P}^{n-1} \times M_n $. Applying the no-name lemma to the $PGL_n $-equivariant dominant rational map $~X \rightarrow Y $ given by $~(A, B) \rightarrow (Ker(A), B) $ (which makes X into a vector bundle over a dense open $PGL_n $-invariant subset of Y), we see that $X//PGL_n $ is rational over $Y//PGL_n $ On the other hand, $Y//PGLn = M_n//PGL_n $ is an affine space. Thus $X//PGL_n $ is rational. The moment I read this I knew how to do this quiver-wise and that it is just another Brauer-Severi type argument so completely inadequate to help settling the genuine matrix-problem. Update on the paper by Esther Beneish : Esther did submit the paper in february.

Half a year
ago, it all started with NeverEndingBooks in
which I set out a rather modest goal :

Why NeverEndingBooks
? We all complain about exaggerated prices of mathematical books from
certain publishers, poor quality of editing and refereeing offered, as
well as far too stringent book-contracts. Rather than lamenting about
this, NeverEndingBooks gives itself one year to learn (and report) about
the many aspects of the book-production cycle and to explore whether an
alternative exists. If at the end of this year we will have produced at
least one book this experiment will be considered a success. If,
however, we find out that it is an impossible task, we will explain
where it all went wrong and why it is better to stick to an established
PublishingHouse and accept its terms.

I assume we did
manage to do it after all as you may check by visiting our storefront :
www.lulu.com/neverendingbooks. However, it all turned out to be
quite different from what I had in mind half a year ago. So, perhaps
it’s time to recap.

Originally, I’d planned to partner-up
with the publisher-on-demand LightningSource but
in the process they did require a VAT-number. In Belgium, the safest
way to get one is to set up a non-profit organization (a VZW as we call
it). But then you have to write down your legal statutes, get them
published in the Moniteur
Belge (at a hefty price) but what really put me off was that you
have to set up a “board of directors” consisting of at least three
people. I don’t mind following a folly but if I have to involve others
I usually pass, so I abandoned the whole idea. Still, I couldn’t help
talking about the VAT-problem and at a certain time there was an idea to
revive a sleeping VZW (=non-profit organization) of the Belgian Mathematical Society, the _MaRC_ (MAthematical Research Centre), the
statutes of which allowed to become a publishing house. But, this
wouldn’t involve just two other people but the whole BMS so I decided
to forget all about it and have a short vacation in France together with
a few (former)PhD-students.

Given
plenty of sun, cheese and whine (not necessarily in that order) sooner
or later we had to talk about _the_ problem. For Raf it was the
first time he heard about it but when we realized I thought one could
easily publish books well under 25 dollars he was immediately interested
and insisted we should set up a board of directors and continue with the
plan. The different roles to play in the board were more or less
self-evident : I had to be the trasurer (given the fact that I was the
only with a secure, though small, income), Geert had to become chairman
(being the only one possessing suits), Raf would be secretary (being the
only one who could write better Flemish than English) and Jan or Stijn would do PR (as they
are the only ones having enough social skills). So, we went back willing
to go through the whole process (at least 3 months) of obtaining a
VAT-number.

But then Raf got so interested in the whole idea
that he explored other possibilities (I think he was more motivated by
the fact that his sister wanted to publish her thesis rather than
anything else) and came up with lulu.com. No legal hassle, no
VAT-numbers, nothing required (or so it seemed). Still, before risking
his sister’s thesis he wanted to check the service out and as it is a
lot easier to take a book lying around rather than write one yourself he
took my version 2 and published it at
Lulu’s (since then this version is nicknamed Rothko@n).

Although I gave him the permission to do so, it didn’t feel right
that people should pay even a small amount for a nicely bound unedited
version 2. So, the last month and a half I’ve been editing and
partially rewriting version 2 and the two volumes are now available! Major changes
are to the 4 middle chapters. There is now chapter 3 “Etale
Technology” which contains all of the etale tricks scattered earlier
in two chapters, chapter 4 ‚”Quiver Representations” collects all the
quiver material (again, scattered throughout the previous version).
Chapter 5 ‚”Semisimple Representations” now includes recent material
such as Raf’s characterization of the smooth locus of Cayley-smooth
orders and our (together with Geert) classification of the central
singularities, and chapter 6 ‚”Nilpotent Representations” now
includes the material on Brauer-Severi varities which was in version 1
but somehow didnt make it to version 2 before.

One of the things I like most about returning from a vacation is to
have an enormous pile of fresh reading : a week's worth of
newspapers, some regular mail and much more email (three quarters junk).
Also before getting into bed after the ride I like to browse through the
arXiv in search for interesting
papers.
This time, the major surprise of my initial survey came
from the newspapers. No, not Bush again, _that_ news was headline
even in France. On the other hand, I didn't hear a word about href="http://news.bbc.co.uk/1/hi/world/europe/3974179.stm"> Theo Van
Gogh being shot and stabbed to death in Amsterdam. I'll come
back to this later.
I'd rather mention the two papers that
somehow stood out during my scan of this week on the arXiv. The first is
Framed quiver moduli,
cohomology, and quantum groups by Markus
Reineke. By the deframing trick, a framed quiver moduli problem is
reduced to an ordinary quiver moduli problem for a dimension vector for
which one of the entries is equal to one, hence in particular, an
indivisible dimension vector. Such quiver problems are far easier to
handle than the divisible ones where everything can at best be reduced
to the classical problem of classifying tuples of $n \\times n$ matrices
up to simultaneous conjugation. Markus deals with the case when the
quiver has no oriented cycles. An important examples of a framed moduli
quiver problem _with_ oriented cycles is the study of
Brauer-Severi varieties of smooth orders. Significant progress on the
description of the fibers in this case is achieved by Raf Bocklandt,
Stijn Symens and Geert Van de Weyer and will (hopefully) be posted soon.

The second paper is Moduli schemes of rank
one Azumaya modules by Norbert Hoffmann and Urich Stuhler which
brings back longforgotten memories of my Ph.D. thesis, 21 years
ago…

[Last time][1] we saw that for $A$ a smooth order with center $R$ the
Brauer-Severi variety $X_A$ is a smooth variety and we have a projective
morphism $X_A \rightarrow \mathbf{max}~R$ This situation is
very similar to that of a desingularization $~X \rightarrow
\mathbf{max}~R$ of the (possibly singular) variety $~\mathbf{max}~R$.
The top variety $~X$ is a smooth variety and there is a Zariski open
subset of $~\mathbf{max}~R$ where the fibers of this map consist of just
one point, or in more bombastic language a $~\mathbb{P}^0$. The only
difference in the case of the Brauer-Severi fibration is that we have a
Zariski open subset of $~\mathbf{max}~R$ (the Azumaya locus of A) where
the fibers of the fibration are isomorphic to $~\mathbb{P}^{n-1}$. In
this way one might view the Brauer-Severi fibration of a smooth order as
a non-commutative or hyper-desingularization of the central variety.
This might provide a way to attack the old problem of construction
desingularizations of quiver-quotients. If $~Q$ is a quiver and $\alpha$
is an indivisible dimension vector (that is, the component dimensions
are coprime) then it is well known (a result due to [Alastair King][2])
that for a generic stability structure $\theta$ the moduli space
$~M^{\theta}(Q,\alpha)$ classifying $\theta$-semistable
$\alpha$-dimensional representations will be a smooth variety (as all
$\theta$-semistables are actually $\theta$-stable) and the fibration
$~M^{\theta}(Q,\alpha) \rightarrow \mathbf{iss}_{\alpha}~Q$ is a
desingularization of the quotient-variety $~\mathbf{iss}_{\alpha}~Q$
classifying isomorphism classes of $\alpha$-dimensional semi-simple
representations. However, if $\alpha$ is not indivisible nobody has
the faintest clue as to how to construct a natural desingularization of
$~\mathbf{iss}_{\alpha}~Q$. Still, we have a perfectly reasonable
hyper-desingularization $~X_{A(Q,\alpha)} \rightarrow
\mathbf{iss}_{\alpha}~Q$ where $~A(Q,\alpha)$ is the corresponding
quiver order, the generic fibers of which are all projective spaces in
case $\alpha$ is the dimension vector of a simple representation of
$~Q$. I conjecture (meaning : I hope) that this Brauer-Severi fibration
contains already a lot of information on a genuine desingularization of
$~\mathbf{iss}_{\alpha}~Q$. One obvious test for this seemingly
crazy conjecture is to study the flat locus of the Brauer-Severi
fibration. If it would contain info about desingularizations one would
expect that the fibration can never be flat in a central singularity! In
other words, we would like that the flat locus of the fibration is
contained in the smooth central locus. This is indeed the case and is a
more or less straightforward application of the proof (due to [Geert Van
de Weyer][3]) of the Popov-conjecture for quiver-quotients (see for
example his Ph.D. thesis [Nullcones of quiver representations][4]).
However, it is in general not true that the flat-locus and central
smooth locus coincide. Sometimes this is because the Brauer-Severi
scheme is a blow-up of the Brauer-Severi of a nicer order. The following
example was worked out together with [Colin Ingalls][5] : Consider the
order $~A = \begin{bmatrix} C[x,y] & C[x,y] \\ (x,y) & C[x,y]
\end{bmatrix}$ which is the quiver order of the quiver setting
$~(Q,\alpha)$ $\xymatrix{\vtx{1} \ar@/^2ex/[rr] \ar@/^1ex/[rr]
& & \vtx{1} \ar@/^2ex/[ll]} $ then the Brauer-Severi fibration
$~X_A \rightarrow \mathbf{iss}_{\alpha}~Q$ is flat everywhere except
over the zero representation where the fiber is $~\mathbb{P}^1 \times
\mathbb{P}^2$. On the other hand, for the order $~B =
\begin{bmatrix} C[x,y] & C[x,y] \\ C[x,y] & C[x,y] \end{bmatrix}$
the Brauer-Severi fibration is flat and $~X_B \simeq \mathbb{A}^2 \times
\mathbb{P}^1$. It turns out that $~X_A$ is a blow-up of $~X_B$ at a
point in the fiber over the zero-representation.

[1]: http://www.neverendingbooks.org/index.php?p=342
[2]: http://www.maths.bath.ac.uk/~masadk/
[3]: http://www.win.ua.ac.be/~gvdwey/
[4]: http://www.win.ua.ac.be/~gvdwey/papers/thesis.pdf
[5]: http://kappa.math.unb.ca/~colin/