# Why is it called a matrix?

Here’s a question for which the answer doesn’t seem to be widely known. Certainly, no one I’ve asked so far has known. They all found the answer interesting though!

Why is a matrix called a matrix?

If we consult www.etymonline.com for the origins of the word matrix we find

matrix (n.)

late 14c., “uterus, womb,” from Old French matrice “womb, uterus,” from Latin matrix (genitive matricis) “pregnant animal,” in Late Latin “womb,” also “source, origin,” from mater (genitive matris) “mother” (see mother (n.1)).

That doesn’t seem to be much help (but does explain the word matriarchy and why surgeons sometimes refer to the matrix). However, surprisingly “womb” is the origin of the word in mathematics. To see this we go back to JJ Sylvester‘s 1850 article Additions to the articles in the September number of this journal, “On a new class of theorems,” and on Pascal’s theorem in which it is used in this context for the first time. He says

For this purpose we must commence, not with a square, but with an oblong arrangement of terms consisting, suppose, of m lines and n columns. This will not in itself represent a determinant, but is, as it were, a Matrix out of which we may form various systems of determinants by fixing upon a number p, and selecting at will p lines and p columns, the squares corresponding to which may be termed determinants of the pth order.

So what is Sylvester talking about here? Well, his interest is in developing what we now think of as Linear Algebra. In particular, he cares about determinants and is thinking of them as arising from a square array of numbers. In the passage above he notes that from an m by n array of numbers — what he calls a Matrix — we can take p columns and rows to create squares from which we can produce determinants. That is, the m by n array *gives birth* to the objects he is interested in.

And so that’s where it comes from!

Now, why is a ring is called a ring? Anyone know?

Wikipedia: “According to Harvey Cohn, Hilbert used the term for a ring that had the property of ‘circling directly back’ to an element of itself. Specifically, in a ring of algebraic integers, all high powers of an algebraic integer can be written as an integral combination of a fixed set of lower powers, and thus the powers ‘cycle back’.”

For a algebraic field we have the following history, The Art of the Infinite – Kaplan on page 37:

“In truly formalist style this collection of axioms wasn’t addressed to one kind of number or another but thought of by its first formulators as characterizing a self-standing whole: a body (Körper) as the German mathematician Richard Dedekind tellingly called it. Schumann might have brooded over this slight to any indwelling spirit, Pascal over the missing will.

What was a body in German became the even less suggestive ‘field’ in English, the formalist point of view being that here was a list of laws and whatever obeyed them—rational, real, or complex numbers, or motions like the rotation of figures on a geometric plane, or chairs or beer mugs—was a field (the lure of abstraction may make mathemati- cians seem like a subspecies on the verge of evolving beyond lives steeped in the senses). Further relations could be deduced from those funda- mental ones, and other relations could be shown not to hold among whatever obeyed them. It was the triumph of the container over the con- tent: the slots stood to one another in specific ways; hence, so must any- thing slotted into them.

To take these laws in all at once, in a continuous sweep as Descartes would have us do, here are the unbroken Tablets of the Law, as delivered to us in 1893 by the equally abstract Heinrich Weber (a man about whom much is, but little more need be, known). They are expressed, only for convenience, in terms of numbers (a pure Formalist would have said: ‘If a, b, and c are elements of the field’, and so on).”

What comes to matrices I must say that the multiplication is often taught in a very unmotivational way, simply as a definition! More natural way would be to consider successive linear maps: (x,y) –> (u,v) and then (u,v) –> (i,j), then the map (x,y) –> (i,j) is the multiplication in matrix form. For some unknown reason (all?) texbooks fail witj this explanation before the definition of matrix multiplication.