The Metonymy of Matrices

I was enchanted with the words metonymy and synecdoche when I learned them in a high school English class. (Well, not so enchanted that I’ve ever been able to remember which one was which with any consistency. Maybe this post will change that. I now have a handy new mnemonic that will help almost no one else: Metonymy is for matrices.)

Synecdoche is a figure of speech where a part is used to represent a whole, or vice versa. “All hands on deck” is a synecdoche: the hands are presumably connected to entire humans, unless you’re watching a horror movie about boats. “Chicago won the World Series” substitutes a larger whole—a city—for a part—one of their baseball teams—though to see Chicago celebrate it, perhaps the entire city really did win. Then again, Chicago is a place containing a lot of things besides people, and the people celebrating the World Series are, well, people, so I guess it’s still synecdoche.

Metonymy is the substitution of an associated word for another term. “Bubbly” for “champagne” might be metonymy, but maybe it’s synecdoche because the bubbles are in the champagne. It’s all very confusing. To top it off, synecdoche is a type of metonymy, so it’s safer to use the term metonymy for all of it. (On the other hand, using synecdoche for all metonymy is synecdoche, which is quite poetic.)

Putting aside the intricacies of word usage, I’ve been thinking about the mathematical metonymy of matrices. When I wrote about the space SO(3) in February, I noted that a matrix is an array of numbers, a description that spectacularly undersells its utility. (It’s like describing Serena Williams as a member of the phylum chordata: not incorrect, but really not the point.)

It’s true: a matrix is an array of numbers. But it’s what you do with the numbers that makes it special. Matrices are used most often in mathematics to represent linear transformations. Most of us learn about simple linear transformations early in our education. The equation y=4x is one example of a linear transformation. It happens to be one that takes in one number, x, and spits out another, y. A linear transformation must preserve addition and scalar multiplication: if you add something to the input, the output should change proportionally, and if you multiply the input by some particular quantity, the output should be multiplied by that quantity.

Linear transformations like y=4x, those that take in one variable and spit out one variable are, frankly, kind of boring. They all look the same as y=4x, except they might have a different number in the place of 4. (4 is just a random number I chose.) But linear transformations can take in a lot more variables than just one and spit out a lot more variables.

For example, suppose we want to define a transformation that takes in two variables and spits out two variables. Let’s say we want to feed it the coordinates (x,y) and get out the numbers (2x+y,x+y). We can define a lot of different functions in similar ways. There’s the function that takes (x,y) to (5x-3y,πx+y), the function that takes (x,y) to (.3x,2y), and a multitude of other functions. Mathematicians developed matrices to handle these functions. The first function can be represented by the matrix

the next by the matrix

the next by

We can use them to represent functions that take in a different number of variables than they spit out. The function that takes (x,y,z) to (z-y,x+2y) gets the matrix

It takes a little bit of practice to get comfortable with applying matrices to vectors, and a little more to get comfortable with multiplying matrices to represent compositions of functions (first performing one function, then performing another function on the output of the first function), but once you do, it is a powerful tool.

As a tool, the matrix is so powerful that it is easy to forget that it is a representation of a function, not a function itself. A matrix truly is just the array of numbers, but I think in this context, most mathematicians are metonymists. (Metonymers? Metonymnistes?) We think of the matrix as the function itself, and it’s easy to lose sight of the fact that it’s only notation. Matrices don’t even have to encode linear transformations. They are used in other contexts in mathematics, too, and restricting our definition to linear transformations can shortchange the other applications (though for my money, the value of the matrix as a way of representing linear transformations dwarfs any other use they have).

After tweeting about how matrices are like Serena Williams, I enjoyed the lively Twitter discussion on the topic of matrices, linear transformations, and teaching linear algebra. Some people were glad they learned linear transformations separately from matrices, some enjoyed the “click” of finally understanding that these matrices they had been struggling with were part of a much bigger picture. I think at some point, there’s some amount of metonymizing that is necessary for people to get to the point where they can comfortably work with matrices as linear transformations and unlock their full power in that context, but it’s probably good that every once in a while we remember that matrices are metonymy.