Dynamics on Graphs 17
We show how the Cartan matrix of a graph leads to geometric structure on the space of populations on that graph. This is the quadratic form first studied by Jacques Tits, and it has an intimate connection with the mutation game on that graph, and in fact also on the dual mutation, or numbers, game on that graph.
This development introduces a rich new direction for geometry: we see that any graph supports a rich metrical geometry which however is not necessarily positive definite. So it becomes important to have a more general approach to metrical geometry -- namely we have to adopt the point of view of Rational Trigonometry.
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0:01
Hello everyone I'm Norman Wildberger thanks for joining me on this adventure we're looking at
0:05
Dynamics on graphs I've introduced the cartoon Matrix of a general graph and show the intimate
0:10
connection between the mutation game and the Dual numbers game today I want to show you that the
0:16
carton Matrix of a graph also opens up a whole new world for us the world of geometry okay and that
0:26
comes about by a certain quadratic form which appears which was discovered by Jacques tits I
0:32
think at first so we'll call it the tits quadratic form now this General topic also plays a big role
0:39
in the representation theory of Lee algebra's and Lead groups I've already mentioned that but
0:43
also in the theory of Quivers so there's also a quiver point of view towards this which perhaps
0:48
we'll talk about at some point okay so I want to illustrate things with an example to be really
0:53
specific and concrete okay I'll leave it to you to try to work through similar examples with other
1:00
relatively small grass so we're going to start with this particular graph that's already very
1:04
familiar to us forward sees one two three four a population is just a function on the vertices
1:09
given by a vector A1 A2 A3 A4 these AIS are integers The adjacency Matrix of the graph is is
1:18
this thing right here and the carton Matrix is by definition two times the identity Matrix minus a
1:25
and I'm denoting minus signs with bars over things just to make things a little bit more compact okay
1:32
so there's the carton Matrix C coming essentially from The adjacency Matrix a of the graph
1:39
okay now Associated to such a graph and populations like this we had this idea of
1:47
mutations which is really at the core of our thinking so there's actually four mutation
1:51
matrices Associated to this graph we've seen that before and here are these four mutation matrices
1:57
consisting of zeros ones and minus ones or One bars okay so S1 is the mutation of the vertex
2:03
one it's represented by this Matrix here more or less the identity Matrix except that the the one
2:10
element in the uh on the diagonal it has been negated and we also add essentially the rest
2:16
of the column of the corresponding vertex in The adjacency Matrix so here's S2 which is again more
2:23
or less the identity except there's a minus one here and we are adding the second column otherwise
2:28
of the adjacency Matrix here's S3 and here's S4 so these are the four mutation matrices which
2:35
basically give us the mutation gain by acting on the right done population or row vectors and
2:43
they also give us pleasantly the Dual mutation game or the numbers game by acting on the left
2:51
on weight or column vectors so let's illustrate that so we'll take S1 okay and here's a general
2:58
population a1a2 a34 we multiply by S1 on the right so there's the vector there's the Matrix okay and
3:05
when we just perform the matrix multiplication we get minus A1 Plus A2 and the A2 A3 A4 haven't
3:12
changed so we indeed have performed the mutation at the vertex a by multiplying by this S1 Matrix
3:19
on the right of our population Vector but correspondingly if we look at the the Dual
3:26
side we have a weight Vector W1 w2w3w4 and we act by the same Matrix S1 there's the same
3:33
Matrix but now on the left then what happens is that W1 entry negates and the W1 is also
3:41
added to the direct adjacent neighbors of w which one of W1 which in this case is W2 so
3:48
here we see the Dual mutation of firing at the vertex one expressed Again by the same Matrix
3:55
so I want to impress somebody this is a very pleasant situation and it's analogous to the same
4:01
kind of thing that we saw when we were talking about permutations so remember that I I wanted to
4:06
Define permutations as matrices and that way those matrices could act on the right on row vectors and
4:13
on the left of column vectors so a given Matrix had a sort of a two-fold interpretation as an
4:19
actual transposition okay it was a bit of an non-standard view on what the symmetric group
4:26
really is so the same kind of thing here the same attitude is being adopted here this Matrix S i
4:32
which we're calling a mutation Matrix actually is effectively a mutation Matrix as well as a dual
4:38
mutation Matrix okay it can act on row vectors on the right or on column vectors on the left
4:47
okay great and now let's just illustrate this with another example so here's our population again
4:52
now let's hit it with S2 on the right there's S2 mutation Matrix corresponding to the second vertex
4:58
and what it does is it changes the A2 to minus A2 and adds some of the Neighbors which in this
5:04
case is all of them and so that's the familiar mutation at the vertex too and correspondingly
5:10
if we had done this on the weight Vector then we couldn't check that we're getting what's uh the
5:16
firing at vertex 2 which negates the value at 2 and adds W2 uh value to all the adjacent Neighbors
5:25
okay and now we come to the main theorem of this lecture right here we're going to call
5:30
this the carton Matrix invariance theorem this is a very important and cool theorem
5:39
it says that for any mutation Matrix S i and in fact for any graph so this is
5:44
very general it would just have a general graph x we consider a mutation Matrix S
5:48
i corresponding to some vertex of that graph then we have that if we take s i
5:57
times C the carton Matrix times S I transpose then we get C back again so s i is a it's
6:07
a matrix as I transpose as it's transpose so we're predicting the product of square
6:13
matrices we're getting another Square Matrix but crucially we're getting back C for any vertex I
6:21
okay um so what does this thing look like so here's an example so let's look at our example
6:27
with our simple graph with four vertices here's the carton Matrix okay it's got two times the
6:33
identity down the middle there and otherwise it's the negation of the adjacency Matrix and
6:39
on the left here we see S1 the mutation Matrix corresponding to the first vertex
6:45
that we've already seen and over here is its transpose which is much the same except that
6:52
this one entry here has moved up here okay if we perform this matrix multiplication well okay
6:59
um this is the same as that and you can multiply this thing here by this thing
7:02
here and you'll get this thing here you can check and then when you do this matrix
7:06
multiplication you get this thing and happily this is the same as the C that we started with
7:15
okay so here is now a major exercise for you to think about and it really deserves if
7:21
you're interested in this subject okay and you're interested in you know really getting into it and
7:25
starting to make some investigations this uh this exercise will reward your thought okay is to find
7:33
a proof of this for a general simple graph x in other words why is this through an explanation for
7:40
why this is true laughs okay so one hint kind of an obvious thing that one's always sort of think
7:49
of first if one doesn't have an initial idea it's just to work through enough explicit examples so
7:55
you can sort of start to see why perhaps it works in general okay that's a good starting point just
8:02
work it out and check first of all that it actually is true okay that's first of all
8:06
giving you confidence in the theorem and perhaps by making all these manipulations we start getting
8:11
a sense of yeah I think perhaps I can see why this can be true in general okay so with that in mind
8:18
here's another example so the same cartoon Matrix C now S2 that's second mutation Matrix there is S2
8:26
there's its transpose we make the computation this time I guess I'm multiplying these two guys first
8:30
times this one and we get this Matrix as a result you can check the computation and we do it now
8:36
indeed gets same okay another hint let me give you another hint about this so another hint that both
8:42
the matrices for S Sub I and C are constructed from essentially two ingredients one of them
8:48
is very simple so the identity Matrix and the other ingredient is really the adjacency Matrix
8:54
directly for the cartoon Matrix or in the case of the mutation of matrices Columns of the adjacency
9:01
Matrix just taking one column at a time and sort of sticking it on uh you know a slightly modified
9:07
identity to get these mutation matrices so it's probably not a bad idea to try to reformulate
9:15
what's going on perhaps in terms of the adjacency Matrix a which is sort of playing the key role
9:21
here isn't it and the identity of course is also involved but the identity is easy to work with
9:27
okay so uh this uh this equation here that we have this theorem it's really a great equation it's
9:36
it's something that geometers immediately uh sit up and take notice because this is suggesting from
9:44
a linear algebra point of view that geometrical structure is being maintained there's some
9:49
geometry going on here because C is a symmetric Matrix okay so we have a symmetric Matrix which
10:00
in Geometry is often associated with a quadratic form which is like sort of a metrical structure
10:07
and this particular combination multiplying by a matrix on the left and it's transpose on the
10:12
right that's the kind of thing that um that suggests that we have some some invariance
10:19
going on that these sis are somehow acting in a pleasant way with respect to this geometry
10:25
so to make that a little bit more explicit Let's uh connect with some linear algebra in
10:30
this context by introducing the tits quadratic form named after Jacques tits prominent French
10:36
mathematician who investigated these things in different directions so for a population
10:43
p on the graph x okay X is a general graph let's define q of P to be this particular combination
10:55
p times c p transpose okay sort of motivated by the form of this uh this theorem here okay it's
11:06
motivated by it but um you know that's that's a very sort of a standard thing if you didn't have
11:13
the C here you just take P times P transpose well that's just giving you what's usually called the
11:18
usual dot product or or inner product or euclidean metrical structure on this n-dimensional space
11:27
if the number of vertices is n so this is a variant on that where we're mixing things up
11:33
by substituting a more general or complicated possibly um symmetric Matrix in the middle of
11:39
the p and the P transpose okay so uh what first of all we'll call this something okay we'll call it
11:49
the quadrants of P suggesting that we're making contact here with rational trigonometry which
11:55
is exactly what we're doing okay so it turns out that rational trigonometry is almost essential as
12:01
we move in this direction to really understand what's what's going on here and to be able to
12:05
tackle the rich geometry that's implicit here okay so here's the the corollary really that um follows
12:13
from our theorem which is that for any vertex I this General graph x and a general population P
12:18
the quadrants of p is the same as the quadrants of P times s i in other words you have a population
12:27
paid can be any kind of population just an integer valued functional vertices you mutate at some
12:34
vertex I to get another population this thing is saying that the quadrants of those two populations
12:43
are going to be equal so the quadrants is being maintained and okay in geometrical
12:53
language that's expressed by the statement that the mutations s i are acting as isometries
13:00
let's prove this okay so let's look at Q of p s i the right hand side so by definition that's p
13:09
times s i times c times a transpose of this combination P S I transpose now when you take
13:17
the transpose of a product of matrices you of course get the product of the transposes but
13:22
in the opposite order so you get S I transpose P transpose and since this is all just Matrix
13:30
multiplication it's associative so we can revamp the brackets we can put brackets around the s i
13:38
the C and the CI transpose which is appearing in the middle here and when we do that then we see
13:44
exactly the expression which our basic theorem the carton Matrix and variance theorem tells us
13:50
is equal to C so we get p c p transpose which of course is the quadrants of P so that's why
13:59
the quadrants of P times s i is equal to the quadrants of p and it directly hinges on this
14:05
important carton Matrix invariance theorem that I've asked you to spend some days or weeks okay
14:13
or maybe hours uh maybe minutes if you're really smart um trying to figure out why that's true
14:22
now um the consequences of this are really quite interesting but let's just um have a
14:28
look at an example okay so suppose that in our example of that vertex for uh graph p is A1 A2
14:36
A3 A4 a general population then the quadrants of P will look like what so well we have to take the
14:41
P which is a row Vector times c times P transpose now the c times P transpose we've seen that before
14:47
that's really what we call W that's the associated weight to P okay whereas W is c times P transfers
14:54
it's another way of thinking about what this is we're just taking the product of the original
14:59
population p with its Associated weight without any intermediate Matrix or just that ordinary sort
15:06
of usual product of those that row vector and the column Vector but here it is explicitly so there's
15:13
the vector P here's the carton Matrix and there's P transpose and well we already sort of know what
15:19
what this combinations can be that's going to be the associated weight and here it is explicitly so
15:24
um yeah 2 a 1 minus A2 there's a second entry the third entry and the fourth entry of the
15:30
corresponding weight just what you get when you multiply these two things here okay and then when
15:34
we multiply this out well we get a bunch of terms okay uh what do we get so if the multiply A1 times
15:41
the first entry plus A2 times the second entry plus A3 times the third entry plus A4 times the
15:46
fourth entry so we're getting a whole bunch of terms which are all of degree two okay so like
15:50
the first row gives two a one squared minus A1 A2 that's those two terms there okay and so I'm
15:57
just writing it out all terms and then there's a bit of collecting of terms and we see that
16:01
the result is 2a1 squared plus two a two squared plus two a three squared plus two a four squared
16:07
minus two a one a two minus two a two a three minus two a two a four minus two a three a four
16:14
now if you have done a linear algebra course before and you've sort of studied a little bit
16:19
about quadratic forms you'll um you'll see that this is a very familiar thing you can more or
16:25
less read this right off just from this Matrix right because you're multiplying on the left
16:28
by a vector and on the right by its transpose so these coefficients that are appearing here
16:34
which are in symmetrical form uses a symmetric Matrix basically it'll allow you to read off
16:39
things right we're saying that we get 2 times the sum of the squares of the the entries
16:46
and then these various cross terms uh give us products like this these two here give us the
16:51
2a1 A2 terms this one here gives us the two uh A2 A3 terms the a2a4 and the A3 A4 terms
17:05
so that's the quadratic form that's the tits quadratic form in this particular case where
17:11
we have this relatively small graph x with just four vertices but you can imagine that yes if you
17:20
start with a bigger graph this is going to be potentially very complicated and it's sort of
17:26
asking us to start considering geometry in higher dimensional spaces because the the dimension of
17:32
the corresponding space here will be the number of vertices of this graph that can be quite High
17:38
and there's a lot of arbitrariness here right there's a lot of possibilities for the nature
17:43
of this quadratic form because you can cook up a lot of um different graphs just a symmetric Matrix
17:50
with the zeros and at ones essentially populating in a certain way okay and now we make another
17:57
little interesting observation from from this case here is that if we start with a Singleton
18:03
population say at the vertex one so if P happens to be one zero zero zero zero then the quadrants
18:09
of p is going to be what okay if the only non-zero term is A1 well all these things here have have
18:16
other terms except for the very first entry so that's the only entry that's not going to be
18:21
killed so we're going to obviously get that the quadrants of p is 2. and now since every root
18:29
of this graph can be obtained from this particular Singleton by applying a sequence of mutations
18:38
we can deduce that we must have that the quadrants of an arbitrary root population Q also is equal
18:47
to two because at every stage when we apply a mutation the previous corollary tells us that
18:53
the quadratic form does not change so since it started as being having value two it's going to
19:00
maintain its value of 2 as we go along so every root of this graph x will have the property that
19:09
you evaluate this complicated quadratic form the tips quadratic form on it you are going to
19:15
get the number two that's already uh you know a little bit um it would be impressive you know
19:24
there's a lot of information for a big Graphics so what we're saying is that all the roots
19:30
in this case lie on a quadric it's actually a quadric surface right given by one equation
19:36
which is homogeneously of degree two in the various variables in this case A1 A2 A3 A4
19:44
now in this case this is a a quadric in a four-dimensional space so we have a
19:49
four-dimensional space spanned by a1a283 A4 and this particular equation here you know if we set
19:56
it equal to two Okay that's going to be a like a sort of a hyper surface in this four-dimensional
20:03
space okay it looks like something rather not at all clear what it looks like but what we're
20:08
saying is that all the roots are going to be vectors which are lying on the surface because
20:15
they all have the property that their quadrants is equal to two okay and this argument holds quite
20:24
generally for a general connected simple graph because we know that by definition a general
20:29
root of a general graph x can be obtained from a single term and it's not too hard to see that if
20:35
you have a Singleton population in other words only one of these entries is is one and all the
20:39
others are zero you're just going to pick up one of the elements on the diagonal the elements of
20:44
a diagonal are all two so you're always going to get two for any graph for any Singleton and
20:49
therefore by extension you're always going to get 2 for any root population for any graph X
21:01
so like maybe your head is swimming uh by now I hope it is a little bit I hope you're thinking
21:04
wow this is totally cool it is totally cool okay because we're bringing in a geometry but
21:09
it's potentially uh much richer geometry than we're used to okay this is not just euclidean
21:16
geometry because these quadratic forms are are potentially much more interesting than the usual
21:21
sort of x squared plus y squared plus Z squared quadratic form that we sort of learned about in
21:26
linear algebra courses okay it's it's um much more complicated than that okay so just to give you an
21:34
idea of what's ahead okay so in three dimensions if we go back to three-dimensional space try to
21:40
visualize quadrics surfaces given by homogeneous quadratic equations well you know there are things
21:46
like ellipsoids okay that's that's a kind of a you know something like x squared plus 2y squared
21:53
plus 3 Z squared equals two you know that's gonna be some kind of ellipsoid or you could get a
21:59
hyperboloid of Revolution perhaps of this kind or you'd get a hyperboloid a revolution perhaps of
22:05
this kind so with two branches okay and what we're saying is that if we were in a three-dimensional
22:12
situation in other words a relatively simple graph with just had three birdses then we would expect
22:18
that the the root populations for that graph could be visualized as points which necessarily lie on
22:26
a quadrics like one of these start so we then have lots of dots okay on on one of these ellipsoids
22:34
or maybe on one of these hyperboloids representing the various root populations
22:39
that we can obtain for this graph x so it's very interesting so not only do we have this quadrant
22:46
but we have this sort of interesting pattern of particular points on this quadric surface
22:55
presumably maybe with some regularity or some some intrinsic interest some beauty
23:01
well actually there's a lot of intrinsic interest a lot of Beauty on the geometry of these things
23:06
and uh so that's going to be really interesting but we're obviously going to have to touch base
23:11
with some geometry and we're gonna have to learn some rational trigonometry otherwise this doesn't
23:17
really get off the ground in any sort of very good way so lots to think about lots to learn.