The inspiration for this column comes not from the epic 1999 film The Matrix, as the title may suggest, but from an episode of Sean Carroll’s Mindscape podcast that I listened to over the summer. The episode features Jacob Barandes, a physicist and philosopher at Harvard who describes—over the course of around three hours—his new mathematical formalism of quantum mechanics.
For those interested, the episode is fantastic, and his formalism contains a lot of exciting mathematics (particularly if you’re interested in stochastic processes). But I want to focus on an aside that occurs at one point in the episode, when Professor Barandes shares how the term “matrix” came to be used in mathematics.
First, I should define what a matrix is mathematically. Very broadly, it is an array of numbers, usually organized into a rectangular format of rows and columns. More particularly, one can add and multiply matrices together, as long as the matrices obey a certain format, and the addition and multiplication follow certain rules.
Even if you haven’t seen the term “matrix” before, you’ve likely used them in various contexts. Introductory coding classes usually cover the storage of data into arrays — if you have a two-dimensional array, and have the computer perform some operations on it, then you are effectively doing matrix mathematics! Matrices are also useful in statistics for quantifying variations between random points in two, three, or higher dimensions and to perform a least-squares fit.
Matrices have been used for thousands of years, first being documented in China to solve systems of linear equations. However, the precise use of the term “matrix” is thanks to English mathematician J.J. Sylvester. It’s an interesting word choice, as matrix means womb in Latin. Sylvester coined it as follows: “I have in previous papers defined a ‘Matrix’ as a rectangular array of terms, out of which different systems of determinants may be engendered from the womb of a common parent.”
A determinant is a specific number associated with matrices, and indeed, one can calculate systems of determinants by removing rows and columns from the “mother” matrix to obtain smaller “child” matrices. Sylvester was in some ways waxing poetic here, which is not a huge surprise, given his lifelong passion for verse!
But I do think the choice of term was incredibly prescient. Sylvester lived prior to the advent of computers, whose amazing capabilities are in no small part due to their effectiveness in performing matrix mathematics. Moreover, matrices play an important role in quantum mechanics (the discussion of this in the Mindscape episode led Professor Barandes to bring up matrix etymology), perhaps our best understanding of physical reality, from which many other key discoveries and technologies have been derived.
I won’t go into the details of this mathematics, but I think this history evinces an important insight, relevant not just for matrices, but discoveries more generally. All of you at Stevens will have the opportunity to discover new things, likely many times over in your time here and your future endeavors. As innovators, you have the exciting experience of being the first to gain an intuition for a novel technology, artwork, or idea.
This is an immense challenge ventures past the frontiers of knowledge require you to make the map, understand things on your own, and then explain those things to others. It is an environment much like the one discovered by taking the red pill in The Matrix. But there is also immense power in this, starting with the language you choose to describe and understand what you’re doing.
So, it’s important to pick your jargon well, and explain what that jargon means to a more general audience, as a mother would explain things to her children. And lastly, it’s equally important to have fun in the process, and believe in yourself, as your parents, guardians, or other mentors believe in you. Happy exploring this academic year!
