A Leap Day 2016 Mathematics A To Z: Matrix

I get to start this week with another request. Today’s Leap Day Mathematics A To Z term is a famous one, and one that I remember terrifying me in the earliest days of high school. The request comes from Gaurish, chief author of the Gaurish4Math blog.

Matrix.

Lewis Carroll didn’t like the matrix. Well, Charles Dodgson, anyway. And it isn’t that he disliked matrices particularly. He believed it was a bad use of a word. “Surely,” he wrote, “[ matrix ] means rather the mould, or form, into which algebraical quantities may be introduced, than an actual assemblage of such quantities”. He might have had etymology on his side. The word meant the place where something was developed, the source of something else. History has outvoted him, and his preferred “block”. The first mathematicians to use the word “matrix” were interested in things derived from the matrix. So for them, the matrix was the source of something else.

What we mean by a matrix is a collection of some number of rows and columns. Inside each individual row and column is some mathematical entity. We call this an element. Elements are almost always real numbers. When they’re not real numbers they’re complex-valued numbers. (I’m sure somebody, somewhere has created matrices with something else as elements. You’ll never see these freaks.)

Matrices work a lot like vectors do. We can add them together. We can multiply them by real- or complex-valued numbers, called scalars. But we can do other things with them. We can define multiplication, at least sometimes. The definition looks like a lot of work, but it represents something useful that way. And for square matrices, ones with equal numbers of rows and columns, we can find other useful stuff. We give that stuff wonderful names like traces and determinants and eigenvalues and eigenvectors and such.

One of the big uses of matrices is to represent a mapping. A matrix can describe how points in a domain map to points in a range. Properly, a matrix made up of real numbers can only describe what are called linear mappings. These are ones that turn the domain into the range by stretching or squeezing down or rotating the whole domain the same amount. A mapping might follow different rules in different regions, but that’s all right. We can write a matrix that approximates the original mapping, at least in some areas. We do this in the same way, and for pretty much the same reason, we can approximate a real and complicated curve with a bunch of straight lines. Or the way we can approximate a complicated surface with a bunch of triangular plates.

We can compound mappings. That is, we can start with a domain and a mapping, and find the image of that domain. We can then use a mapping again and find the image of the image of that domain. The matrix that describes this mapping-of-a-mapping is the one you get by multiplying the matrix of the first mapping and the matrix of the second mapping together. This is why we define matrix multiplication the odd way we do. Mapping are that useful, and matrices are that tied to them.

I wrote about some of the uses of matrices in a Set Tour essay. That was based on a use of matrices in physics. We can describe the changing of a physical system with a mapping. And we can understand equilibriums, states where a system doesn’t change, by looking at the matrix that approximates what the mapping does near but not exactly on the equilibrium.

But there are other uses of matrices. Many of them have nothing to do with mappings or physical systems or anything. For example, we have graph theory. A graph, here, means a bunch of points, “vertices”, connected by curves, “edges”. Many interesting properties of graphs depend on how many other vertices each vertex is connected to. And this is well-represented by a matrix. Index your vertices. Then create a matrix. If vertex number 1 connects to vertex number 2, put a ‘1’ in the first row, second column. If vertex number 1 connects to vertex number 3, put a ‘1’ in the first row, third column. If vertex number 2 isn’t connected to vertex number 3, put a ‘0’ in the second row, third column. And so on.

We don’t have to use ones and zeroes. A “network” is a kind of graph where there’s some cost associated with each edge. We can put that cost, that number, into the matrix. Studying the matrix of a graph or network can tell us things that aren’t obvious from looking at the drawing.

Lewis Carroll Tries Changing The Way You See Trigonometry

1861 The Athenaeum magazine reviews Charles Dodgson's suggested new symbols for the six basic trig functions. http://t.co/0xUuDR3XPX

— On This Day in Math (@OnThisDayinMath) July 27, 2015

Today’s On This Day In Math tweet was well-timed. I’d recently read Robin Wilson’s Lewis Carroll In Numberland: His Fantastical Mathematical Logical Life. It’s a biography centered around Charles Dodgson’s mathematical work. It shouldn’t surprise you that he was fascinated with logic, and wrote texts — and logic games — that crackle with humor. People who write logic texts have a great advantage on other mathematicians (or philosophers). Almost any of their examples can be presented as a classically structured joke. Vector calculus isn’t so welcoming. But Carroll was good at logic-joke writing.

Developing good notation was one of Dodgson/Carroll’s ongoing efforts, though. I’m not aware of any of his symbols that have got general adoption. But he put forth some interesting symbols to denote the sine and cosine and other trigonometric functions. In 1861, the magazine The Athanaeum reviewed one of his books, with its new symbols for the basic trigonometric functions. (The link shows off all these symbols.) The reviewer was unconvinced, apparently.

I confess that I am, too, but mostly on typographical grounds. It is very easy to write or type out “sin θ” and get something that makes one think of the sine of angle θ. And I’m biased by familiarity, after all. But Carroll’s symbols have a certain appeal. I wonder if they would help people learning the functions keep straight what each one means.

The basic element of the symbols is a half-circle. The sine is denoted by the half-circle above the center, with a vertical line in the middle of that. So it looks a bit like an Art Deco ‘E’ fell over. The cosine is denoted by the half circle above the center, but with a horizontal line underneath. It’s as if someone started drawing Chad and got bored and wandered off. The tangent gets the same half-circle again, with a horizontal line on top of the arc, literally tangent to the circle.

There’s a subtle brilliance to this. One of the ordinary ways to think of trigonometric functions is to imagine a circle with radius 1 that’s centered on the origin. That is, its center has x-coordinate 0 and y-coordinate 0. And we imagine drawing the line that starts at the origin, and that is off at an angle θ from the positive x-axis. (That is, the line that starts at the origin and goes off to the right. That’s the direction where the x-coordinate of points is increasing and the y-coordinate is always zero.) (Yes, yes, these are line segments, or rays, rather than lines. Let it pass.)

The sine of the angle θ is also going to be the y-coordinate of the point where the line crosses the unit circle. That is, it’s the vertical coordinate of that point. So using a vertical line touching a semicircle to suggest the sine represents visually one thing that the sine means. And the cosine of the angle θ is going to be the x-coordinate of the point where the line crosses the unit circle. So representing the cosine with a horizontal line and a semicircle again underlines one of its meanings. And, for that matter, the line might serve as a reminder to someone that the sine of a right angle will be 1, while the cosine of an angle of zero is 1.

The tangent has a more abstract interpretation. But a line that comes up to and just touches a curve at a single point is, literally, a tangent line. This might not help one remember any useful values for the tangent. (That the tangent of zero is zero, the tangent of half a right angle is 1, the tangent of a right angle is undefined). But it’s still a guide to what things mean.

The cotangent is just the tangent upside-down. Literally; it’s the lower half of a circle, with a horizontal line touching it at its lowest point. That’s not too bad a symbol, actually. The cotangent of an angle is the reciprocal of the tangent of an angle. So making its symbol be the tangent flipped over is mnemonic.

The secant and cosecant are worse symbols, it must be admitted. The secant of an angle is the reciprocal of the cosine of the angle, and the cosecant is the reciprocal of the sine. As far as I can tell they’re mostly used because it’s hard to typeset . And to write instead would be confusing as that’s often used for the inverse sine, or arcsine, function. I don’t think these symbols help matters any. I’m surprised Carroll didn’t just flip over the cosine and sine symbols, the way he did with the cotangent.

The versed sine function is one that I got through high school without hearing about. I imagine you have too. The versed sine, or the versine, of an angle is equal to one minus the cosine of the angle. Why do we need such a thing? … Computational convenience is the best answer I can find. It turns up naturally if you’re trying to work out the distance between points on the surface of a sphere, so navigators needed to know it.

And if we need to work with small angles, then this can be more computationally stable than the cosine is. The cosine of a small angle is close to 1, and the difference between 1 and the cosine, if you need such a thing, may be lost to roundoff error. But the versed sine … well, it will be the same small number. But the table of versed sines you have to refer to will list more digits. There’s a difference between working out “1 – 0.9999” and working with “0.0001473”, if you need three digits of accuracy.

But now we don’t need printed tables of trigonometric functions to get three (or many more) digits of accuracy. So we can afford to forget the versed sine ever existed. I learn (through Wikipedia) that there are also functions called versed cosines, coversed sines, hacoversed cosines, and excosecants, among others. These names have a wonderful melody and are almost poems by themselves. Just the same I’m glad I don’t have to remember what they all are.

Carroll’s notation just replaces the “sin” or “cos” or “tan” with these symbols, so you would have the half-circle and the line followed by θ or whatever variable you used for the angle. So the symbols don’t save any space on the line. They take fewer pen strokes to write, just two for each symbol. Writing the symbols out by hand takes three or four (or for cosecant, as many as five), unless you’re writing in cursive. They’re still probably faster than the truncated words, though. So I don’t know why precisely the symbols didn’t take hold. I suppose part is that people were probably used to writing “sin θ”. And typesetters already got enough hazard pay dealing with mathematicians and their need for specialized symbols. Why add in another half-dozen or more specialized bits of type for something everyone’s already got along without?

Still, I think there might be some use in these as symbols for mathematicians in training. I’d be interested to know how they serve people just learning trigonometry.

The Alice in Wonderland Sesquicentennial

150th anniversary of Lewis Carroll's Alice in Wonderland. My review of a bio focusing on his more mathematical work. http://t.co/9LjViHpM0N

— John Allen Paulos (@JohnAllenPaulos) June 28, 2015

I did not realize it was the 150th anniversary of the publication of Alice in Wonderland, which is probably the best-liked piece of writing by any mathematician. At least it’s the only one I can think of that’s clearly inspired a Betty Boop cartoon. I’ve had cause to talk about Carroll’s writing about logic and some other topics in the past. (One was just a day short of three years ago, by chance.)

As mentioned in his tweet John Allen Paulos reviewed a book entirely about Lewis Carroll/Charles Dodgson’s mathematical and logic writing. I was unaware of the book before, but am interested now.

Lewis Carroll and my Playing With Universes

I wanted to explain what’s going on that my little toy universes with three kinds of elements changing to one another keep settling down to steady and unchanging distributions of stuff. I can’t figure a way to do that other than to introduce some actual mathematics notation, and I’m aware that people often find that sort of thing off-putting, or terrifying, or at the very least unnerving.

There’s fair reason to: the entire point of notation is to write down a lot of information in a way that’s compact or easy to manipulate. Using it at all assumes that the writer, and the reader, are familiar with enough of the background that they don’t have to have it explained at each reference. To someone who isn’t familiar with the topic, then, the notation looks like symbols written down without context and without explanation. It’s much like wandering into an Internet forum where all the local acronyms are unfamiliar, the in-jokes are heavy on the ground, and for some reason nobody actually spells out Dave Barry’s name in full.

Let me start by looking at the descriptions of my toy universe: it’s made up of a certain amount of hydrogen, a certain amount of iron, and a certain amount of uranium. Since I’m not trying to describe, like, where these elements are or how they assemble into toy stars or anything like that, I can describe everything that I find interesting about this universe with three numbers. I had written those out as “40% hydrogen, 35% iron, 25% uranium”, for example, or “10% hydrogen, 60% iron, 30% uranium”, or whatever the combination happens to be. If I write the elements in the same order each time, though, I don’t really need to add “hydrogen” and “iron” and “uranium” after the numbers, and if I’m always looking at percentages I don’t even need to add the percent symbol. I can just list the numbers and let the “percent hydrogen” or “percent iron” or “percent uranium” be implicit: “40, 35, 25”, for one universe’s distribution, or “10, 60, 30” for another.

Letting the position of where a number is written carry information is a neat and easy way to save effort, and when you notice what’s happening you realize it’s done all the time: it’s how writing the date as “7/27/14” makes any sense, or how a sports scoreboard might compactly describe the course of the game:

0 1 0   1 2 0   0 0 4   8 13 1
2 0 0   4 0 0   0 0 1   7 15 0

To use the notation you need to understand how the position encodes information. “7/27/14” doesn’t make sense unless you know the first number is the month, the second the day within the month, and the third the year in the current century, and that there’s an equally strong convention putting the day within the month first and the month in the year second presents hazards when the information is ambiguous. Reading the box score requires knowing the top row reflects the performance of the visitor’s team, the bottom row the home team, and the first nine columns count the runs by each team in each inning, while the last three columns are the total count of runs, hits, and errors by that row’s team.

When you put together the numbers describing something into a rectangular grid, that’s termed a matrix of numbers. The box score for that imaginary baseball game is obviously one, but it’s also a matrix if I just write the numbers describing my toy universe in a row, or a column:

40
35
25

or

10
60
30

If a matrix has just the one column, it’s often called a vector. If a matrix has the same number of rows as it has columns, it’s called a square matrix. Matrices and vectors are also usually written with either straight brackets or curled parentheses around them, left and right, but that’s annoying to do in HTML so please just pretend.

The matrix as mathematicians know it today got put into a logically rigorous form around 1850 largely by the work of James Joseph Sylvester and Arthur Cayley, leading British mathematicians who also spent time teaching in the United States. Both are fascinating people, Sylvester for his love of poetry and language and for an alleged incident while briefly teaching at the University of Virginia which the MacTutor archive of mathematician biographies, citing L S Feuer, describes so: “A student who had been reading a newspaper in one of Sylvester’s lectures insulted him and Sylvester struck him with a sword stick. The student collapsed in shock and Sylvester believed (wrongly) that he had killed him. He fled to New York where one os his elder brothers was living.” MacTutor goes on to give reasons why this story may be somewhat distorted, although it does suggest one solution to the problem of students watching their phones in class.

Cayley, meanwhile, competes with Leonhard Euler for prolific range in a mathematician. MacTutor cites him as having at least nine hundred published papers, covering pretty much all of modern mathematics, including work that would underlie quantum mechanics and non-Euclidean geometry. He wrote about 250 papers in the fourteen years he was working as a lawyer, which would by itself have made him a prolific mathematician. If you need to bluff your way through a mathematical conversation, saying “Cayley” and following it with any random noun will probably allow you to pass.

MathWorld mentions, to my delight, that Lewis Carroll, in his secret guise as Charles Dodgson, came in to the world of matrices in 1867 with an objection to the very word. In writing about them, Dodgson said, “”I am aware that the word `Matrix’ is already in use to express the very meaning for which I use the word `Block’; but surely the former word means rather the mould, or form, into which algebraical quantities may be introduced, than an actual assemblage of such quantities”. He’s got a fair point, really, but there wasn’t much to be done in 1867 to change the word, and it’s only gotten more entrenched since then.

Reading the Comics, July 14, 2012

I hope everyone’s been well. I was on honeymoon the last several weeks and I’ve finally got back to my home continent and new home so I’ll try to catch up on the mathematics-themed comics first and then plunge into new mathematics content. I’m splitting that up into at least two pieces since the comics assembled into a pretty big pile while I was out. And first, I want to offer the link to the July 2 Willy and Ethel, by Joe Martin, since even though I offered it last time I didn’t have a reasonably permanent URL for it.

Why You Failed Your Logic Test

An interesting parallel’s struck me between nonexistent things and the dead: you can say anything you want about them. At least in United States law it’s not possible to libel the dead, since they can’t be hurt by any loss of reputation. That parallel doesn’t lead me anywhere obviously interesting, but I’ll take it anyway. At least it lets me start this discussion without too closely recapitulating the previous essay. The important thing is that at least in a logic class, if I say, “all the coins in this purse are my property”, as Lewis Carroll suggested, I’m asserting something I say is true without claiming that there are any coins in there. Further, I could also just as easily said “all the coins in this purse are not my property” and made as true a statement, as long as there aren’t any coins there.

Getting This Existence Thing Straight

Midway through “What Lewis Carroll Says Exists That I Don’t” I put forth an example of claiming a property belongs to something which clearly doesn’t exist. The problem — and Carroll was writing this bit, in Symbolic Logic, at a time when it hadn’t reached the current conclusion — is about logical propositions. If you assert it to be true that, “All (something) have (a given property)”, are you making the assertion that the thing exists? Carroll gave the example of “All the sovereigns in that purse are made of gold” and “all the sovereigns in that purse are my property”, leading to the conclusion, “some of my property is made of gold”, and pointing out that if you put that syllogism up to anyone and asked if she thought you were asserting there were sovereigns in that purse, she’d say of course. Carroll has got the way normal people talk in normal conversations on his side here. Put that syllogism before anyone and point out that nowhere is it asserted that there are any coins in the purse and you’ll get a vaguely annoyed response, like when the last chapter of a murder cozy legalistically parses all the alibis until nothing makes sense.

What I Call Some Impossible Logic Problems

I’m sorry to go another day without following up the essay I meant to follow up, but it’s been a frantically busy week on a frantically busy month and something has to give somewhere. But before I return the Symbolic Logic book to the library — Project Gutenberg has the first part of it, but the second is soundly in copyright, I would expect (its first publication in a recognizable form was in the 1970s) — I wanted to pick some more stuff out of the second part.

When To Run For The Train

I mean to return to the subject brought up Monday, about the properties of things that don’t exist, since as BunnyHugger noted I cheated in talking briefly about what properties they have or don’t have. But I wanted to bring up a nice syllogism whose analysis I’d alluded to a couple weeks back, and which it turns out I’d remembered wrong, in details but not in substance.

What Lewis Carroll Says Exists That I Don’t

I borrowed from the library Symbolic Logic, a collection of an elementary textbook — intended for children, and more fun than usual because of that — on logic by Lewis Carroll, combined with notes and manuscript pages which William Warren Bartley III found toward the second volume in the series. The first part is particularly nice since it’s text that not only was finished in Carroll’s life but went through several editions so he could improve the unclear parts. In case I do get to teaching a new logic course I’ll have to plunder it for examples as well as for this rather nice visual representation Carroll used for sorting out what was implied by a set of propositions regard “All (something) are (something else)” and “Some (something) are (this)” and “No (something) are (whatnot)”. It’s not quite Venn diagrams, although you can see them from there. Oddly, Carroll apparently couldn’t; there’s a rather amusing bit in the second volume where Carroll makes Venn diagrams out to be silly because you can make them terribly complicated.

A Night In Wonderland

haggisthesheep here offers a pleasant report about a math-oriented event in the National Museum of Scotland recently. The theme was “A Night In Wonderland”, which is probably almost inevitable, since mathematicians really, really like that (a) people have heard of Lewis Carrol, (b) he can be fairly described as a mathematician, and (c) people aren’t scared of mathematics when it’s presented as Lewis Carroll clowning around.

I recall flipping through one of his logic books and finding a delightful demonstration of how the conclusion may be true while the arguments are not — I believe it was “if a person is late for a train, then he will be running” and “the man is running”, doesn’t prove he’s late for the train, because he might be being chased by a tiger. A good tiger chase livens up any discussion of logic.

Knot your average sheep...

On 18th May I was lucky enough to get involved with my first RBS Museum Lates at the National Museum of Scotland. These events happen about 3 times a year and are a chance for the (over 18) public to come back into the museum after hours and to get cosy with the exhibits with a cocktail and live band. It’s also a chance for science (and arts!) communicators like me to run an activity and get some surreptitious education into the evening.

The theme for this month’s Museum Late was “A Night in Wonderland”, so there were lots of top hats, white rabbits and red queens! (See lots of photos of the event on the Museum’s Flickr page.) Knowing that Lewis Carroll (real name Charles Dodgson) was a mathematician and logician as well as nonsense-poem writer, it seemed wrong for there not to be a mathematical component to…

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