Transcriber: Emma Gon Reviewer: Kwangmin Lee
I'm going to take you back to class,
to school, to algebra,
and that's probably the last thing
that you wanted to do in a TED talk today.
Maybe you remember in algebra class that you learned
how to solve small systems of equations
in variables X, Y, Z,
and maybe they didn't tell you exactly
what these X, Y, Z were,
and you were left with the impression
that these equations were not particularly interesting
nor particularly beautiful.
But actually equations like these
are the very core of science and engineering,
and all the algebra around them.
And because each of these equations
defines a relationship between these variables X, Y, and Z,
and relationships and connections are all around us.
Now, for that reason, we love them,
and I want to share some of that love with you today.
Maybe you remember how to solve these equations,
you express most of these variables in terms of one or another, here Y
and after some manipulations, you get a final equation for Y
that you can then solve that gives you back X and then Z.
That works really well for these small systems of equations.
But perhaps you had these nightmares
of going into class one day,
and your teacher writing down on the board
this test of equations and saying,
"Go ahead and solve them."
Now the equations that I work with,
have thousands if not millions of variables,
and obviously we would really need a very big piece of paper,
and a lot of patience to solve in the way you were taught.
So, of course, we don't.
Instead we use computer programs.
Now, before we can use these computer programs though,
we need to reorder these equations a little bit.
So we're going to write them really really neatly
under each other and where variables are missing,
for example, Z in that first equation
we add it but we multiply it with a zero.
And then we explicity write all the coefficients in front of it.
Write everything in terms of addition like this.
It seems a little bit silly, why would you do that?
But now you see, every equation looks the same.
Something times X plus something times Y
plus something time Z is something else.
And so we don't have to write X, Y and Z all the time,
we just remember the order in which they occurr, right?
And we store these X, Y and Z
in a little skinny table that we call a vector,
like this, and then we store these coefficients
the ones and the zeros and here also a minus one
in a separate table, like -
I will show you in a minute, here.
So now the system of equations that I have
is really just a table of coefficients like this
and then these little vectors with all these unknowns.
Now this table of coefficients we call
the matrix.
And the matrix is so famous,
they even made a few movies after it. (Laughter)
And like Morpheus says in this movie,
"The matrix is everywhere.
It's all around us."
Even now in this very room.
So I want to give you a few examples
of where matrices occur.
And as first example I am going to take you
to the San Francisco Bay.
So here in the San Francisco Bay,
some of my colleagues designed a really nice computer simulation
of the tidal flows going in and out the bay.
And this is a really interesting simulation
that can show you for example,
salinity gradients in the bay
or maybe surface velocities that are good
and useful for the America's Cup
which is exactly what they did last year.
Now the flow is much too complex to understand
and computer velocity in every single point.
So instead we want to be computing
the velocities in a set of points
distributed throughout the domain
and here the points are the vertices of these triangles.
Now through the laws of physics,
we can relay the velocity in each of these vertices
to velocities in neighboring points.
There is a relationship between them.
If there is a relationship, there must be an equation,
and if I have a whole bunch of these equations,
what do I get?
The matrix.
Now, if I write down this matrix it will be extremely large,
and it will have a lot of numbers in it, so I don't do that,
instead for every non-zero in this matrix
I put a little blue dot, right?
And then what I get is a matrix like this
that we can actually look at from afar
and then you can see structure in this.
Now we use computer programs to create matrices like this
or visuals of matrices like this
and these are called spy plots,
and sometimes these computer programs
they have little Easter eggs in them.
And so, one of these programs
when you type the command, spy
you see this. (Laughter)
Now we can do the same thing in your body
I don't know if you realize this,
but your body has matrices in [it].
And now I'm going to take you to a simulation
of my colleagues in Med USC in San Diego
of blood flow through the aorta.
Now this blood flow model was created using CT scanning of the aorta itself.
And besides the blood flow
it will also give you really interesting information about wall shear stresses
that are important for blood clotting
that you want to know about for bypass operations.
And again, the velocities and the wall shear stress
are computed in points distributed like this.
And the matrix that comes out
has a really interesting structure, too.
Now, search engines use matrices that tell them
which words occur in which website.
And I want to take four randomly chosen words:
Stanford, beating, California,
and the word, egg, right?
And five websites that have one or more of these words in them.
And so, there is a website over cooking,
over TEDxStanford, the Berkeley website,
Now what the search engine does
it creates a beautiful table,
where each row of the table corresponds to a word,
and each column of the table corresponds to a web page.
And if a word occurs in the web page then you put a little one
and if it doesn't, then you put a zero
And what do you get out of this? A matrix.
Now in reality these matrices are billions of web pages long
and millions of words thick.
And you need to be a pretty good mathematician
in order to build a good search engine.
So, you know now that this is a good field to be in
to earn a little bit of money.
So here are those matrices.
Now a very variation on this theme
is shown in the next slide of a matrix
and here every column, every row corresponds to a text document,
and there is a little blue dot if these two documents
have a lot of words in common.
So this is a connection matrix for 20 000 documents from the Classics.
We can do the same with web pages,
we can put web pages both along the rows and the columns,
and put a little blue dot if there is a hyperlink from one web page to another.
And what you are looking at here now
is the Stanford and Berkeley web domains.
With Stanford nicely clustered, Berkeley nicely clustered,
but there is communication between these two,
because there are some blue dots in the other side as well,
which is actually kind of surprising
since we keep beating them. (Laughter)
But they still want to communicate with us.
Now, matrices are also in your brain.
Here you are looking at white matter fibers
that connect gray matter regions in your brain.
And looking at this, I could create a matrix
that has gray matter regions in your brain
along the rows and the columns
and show connectivity between these regions.
So, here you can stare this matrix for a long time
and understand how your brain is all connected up.
So matrices are everywhere,
they model systems of equations,
they're in your brain, they're in the search engine,
and as mathematicians, we work for them daily,
and we really love them.
And these matrices, they have personalities for us.
So, when I prepared for this talk,
I asked my students, what are your favorite matrices?
And here are a couple that I wanted to share with you.
So we have some matrices that are sparse with a lot of zeros,
and then in the top right, the matrices that are symmetric,
and symmetry, as we know, is always signifying beauty,
so we love those.
And then the matrices that are symmetric and sparse,
and sort of bended like this
so they are really fantastic.
And then there are other matrices which I really like,
a sort of blockage structures.
But the very winner of this competition
was the matrix we called the Toeplitz matrix,
and that comes up a lot in signal processing.
And it may not look like much to you,
every diagonal in this matrix has a constant number
and it makes it much easier to work with.
Now what about the nastiest matrices
and here is the one that we hate the most.
There are two elements
that are very different in size and makes it really hard to work with.
We call them ill-conditioned.
But the worst matrices in the whole world
are very very large matrices that are ill-conditioned,
and to be able to work with those,
we need to write specialized computer programs,
and so there are millions of lines of codes everywhere.
Now the people wrote [them] in order to manipulate
these very large matrices.
For example, the matrix you heard about this morning
for climate models, it is a very nasty matrix indeed.
And these lines of code, they are behind of a lot of the simulation tools
that you will see out there
and that people use in science and engineering.
Now, I said, I was going to talk about the beauty of math,
and maybe so far, you're not super excited
to say, we've seen some blue dots from the screen
that's not very beautiful.
So let me show you how we put beauty into mathematics and algebra.
Here is a matrix and I'm going to associate
a number with each row and column, one through four, right?
And every number is now going to be related
to a note or a little ball.
There are going to be four of them, one, two, three, four.
And now whenever I have a non-zero
in the matrix, for example, at positions
in the first row, positions one and three,
it means there is a connection between one and one
and one and three.
Now one and one doesn't show up,
but one and three does.
So the next row gives me a connection between two and four,
the next row gives me an additional connection between three and four.
And now I draw a connection, the last one,
between one and four.
Now we have a little graph that comes out,
and that looks much more pleasing than just a table with numbers.
But you can just imagine that if I have a really big matrix
with lots of ones and zeros what a mess that would be.
All these little balls and all these little links between them
So here is the trick.
Whenever there is a link between these two balls
we imagine there is a spring that pulls these balls together.
But at the same time, we give the balls also repelling charges,
so that when they're not connected by a link
they repel each other, push each other away,
and then we just let things go, and sort themselves out.
And what looked to be a mess, just a simple matrix that was messy
is turning into a beautiful structure.
It just finds the minimum energy,
and the outcome is this fantastic geometrical figure.
And when we study them, we can actually see
some of the properties of the physics,
even coming out in these pictures.
Now we can apply those to lots of different matrices.
And I want to share some of the prettiest ones.
Here is people and the web pages they like.
Modelling of a lung, the matrix corresponding to that.
Financial portfolio analysis.
Shallow water models, estuary flows and so on.
The Standford web.
MRI modelling.
Analog circuits... makes your hair stand up.
Tidal flow models.
And my very favorite one, is called the galaxy
that tells you how the catalogs and sub catalogs
in the Library Congress are all connected together.
So, that math that you learned at high school
that these equations that you didn't like so much,
that's behind everything.
It is beautiful, it's omnipresent,
It's everywhere and who knew
you could make such pretty pictures.
Thank you very much. (Applause)