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 explicitly 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 times 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 occur, 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. 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. 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.