# Source of term "multiplication" in matrix multiplication

W

#### William Hughes

Recently, the question was asked in comp.lang.c
why matrix multiplication (a rather complex operation
not obviously related to ordinary mulitplication)
is known as multiplication. Hypotheses have included
the fact that matrix multiplication of nxn matriices
corresponds to the "multiplication" operation in the
ring of nxn matrices, and the fact that matrix
multiplication corresponds to composition of linear
transforms and composition is often termed
multiplication. However, no hard evidence has
been presented.

Does anyone know how the term arose?

- William Hughes

C

#### Chip Eastham

Recently, the question was asked in comp.lang.c
why matrix multiplication (a rather complex operation
not obviously related to ordinary mulitplication)
is known as multiplication.  Hypotheses have included
the fact that matrix multiplication of nxn matriices
corresponds to the "multiplication" operation in the
ring of nxn matrices, and the fact that matrix
multiplication corresponds to composition of linear
transforms and composition is often termed
multiplication.  However, no hard evidence has
been presented.

Does anyone know how the term arose?

- William Hughes

The name is apt for the reasons you mentioned and
others (such as distributivity over matrix addition,
consistency with the computation of determinants,
contrast with scalar multiplication, and its role
in abstracting out coefficients in a system of
linear equations into a single matrix "coefficient").
If the name were not apt, a replacement would be
found.

the term? I recall a thread in this newsgroup
some while back that touched on the history as
regards the priority of matrix notation vs.
determinant notation. If you like I can dig
for it.

regards, chip

A

#### A N Niel

If you like I can dig for it.

Wow. Instead of just guessing at the answer, looking up a reference?
What a novel idea!

B

#### BillyGates

A said:
Wow. Instead of just guessing at the answer, looking up a reference?
What a novel idea!

A reference of someone guessing? Your right... that is novel!

B

#### Ben Bacarisse

William Hughes said:
Recently, the question was asked in comp.lang.c
why matrix multiplication (a rather complex operation
not obviously related to ordinary mulitplication)
is known as multiplication. Hypotheses have included
the fact that matrix multiplication of nxn matriices
corresponds to the "multiplication" operation in the
ring of nxn matrices, and the fact that matrix
multiplication corresponds to composition of linear
transforms and composition is often termed
multiplication. However, no hard evidence has
been presented.

Does anyone know how the term arose?

It seems to be generally agreed that the term 'matrix' was coined by
Sylvester in 1950 as "an oblong arrangement of terms" but he never
viewed matrices as objects in their own right. He was only concerned
with the determinants that they give rise to, not the matrices
themselves.

His friend Cayley seems to have been the first to see that these
objects can have an algebra of their own. In 1858 he published "A
Memoir on the Theory of Matrices" (in the Philosophical Transactions
of the Royal Society of London). In this paper he says that they may
be "multiplied or compounded together" and gives the usual rule though
he is obviously wary of non-square matrices since he leaves a full
discussion of their "composition" to the end of the paper.

He often uses the term "multiplied or compounded" (and sometimes
"composed") but he does use the term "multiplication of matrices" a
few times without qualifying it in any way. What we now call pre- and
post-multiplication, he calls "compounded as the first or second
component".

Interestingly, when he addresses non-square matrices specifically he
switches to using the term "composition" exclusively. I suspect that
the term "multiplication" carried (at the time) too much baggage to be
associated with such an operation.

S

#### Stefan Ram

William Hughes said:
Does anyone know how the term arose?

We can start observing that there is the »natural«
So, therefore, this is called »addition«.

I assume that the algebraic laws for matrix
multiplication then relate it to matrix addition
in the way multiplication is related to addition
in a ring (which is the most general algebraic
structure with a multiplication and commutative
addition that I know). So, therefore, it then would
be »the multiplication in the ring of matrices«.

Indeed, I now have confirmed via Wikipedia that
the matrices over a ring are a ring themselves.

B

#### Ben Bacarisse

We can start observing that there is the Â»naturalÂ«
So, therefore, this is called Â»additionÂ«.

I assume that the algebraic laws for matrix
multiplication then relate it to matrix addition
in the way multiplication is related to addition
in a ring (which is the most general algebraic
structure with a multiplication and commutative
addition that I know). So, therefore, it then would
be Â»the multiplication in the ring of matricesÂ«.

Indeed, I now have confirmed via Wikipedia that
the matrices over a ring are a ring themselves.

s/matrices/NxN matrices/ (i.e. what you say is only true of square
matrices of a particular size).

Overall, the above sounds like a post-justification rather than a
probably explanation of the term. Ring theory started with
Dedekind in about 1870 and it was not until the early 20th century
that rings were unified by axiomatising the abstract structure
they all share.

I get the feeling that Cayley called his composition operation
"multiplication" because he observed that, for square matrices, there
is both a zero (additive and multiplicative) and a multiplicative
identity. It seems unlikely that the term was chosen because of some
deeper algebraic understanding. For example, despite listing 58
properties and theorems about matrices[1] the paper I cited does not
include that fact that multiplication distributes over addition.

[1] Including an unproved version of what came to be called the
Cayley-Hamilton theorem: that a square matrix satisfies its own
characteristic equation p(t) = det(tI - A).

B

#### BillyGates

William said:
Recently, the question was asked in comp.lang.c
why matrix multiplication (a rather complex operation
not obviously related to ordinary mulitplication)
is known as multiplication. Hypotheses have included
the fact that matrix multiplication of nxn matriices
corresponds to the "multiplication" operation in the
ring of nxn matrices, and the fact that matrix
multiplication corresponds to composition of linear
transforms and composition is often termed
multiplication. However, no hard evidence has
been presented.

Does anyone know how the term arose?

- William Hughes

I'm curious as to why it matters. Can you give me a good argument why
knowing the true etymology has any real significance in mathematics? We can
propose several likely reasons that are all logically coherent. Is it really
important to know which one is historically true? We can imagine that any of
the logical reasons could have be used in any alternate history and being
"wrong" here has no negative impact on our intellectual evolution?

C

#### Chip Eastham

We can start observing that there is the »natural«
So, therefore, this is called »addition«.
I assume that the algebraic laws for matrix
multiplication then relate it to matrix addition
in the way multiplication is related to addition
in a ring (which is the most general algebraic
structure with a multiplication and commutative
addition that I know). So, therefore, it then would
be »the multiplication in the ring of matrices«.
Indeed, I now have confirmed via Wikipedia that
the matrices over a ring are a ring themselves.

s/matrices/NxN matrices/ (i.e. what you say is only true of square
matrices of a particular size).

Overall, the above sounds like a post-justification rather than a
probably explanation of the term.  Ring theory started with
Dedekind in about 1870 and it was not until the early 20th century
that rings were unified by axiomatising the abstract structure
they all share.

I get the feeling that Cayley called his composition operation
"multiplication" because he observed that, for square matrices, there
is both a zero (additive and multiplicative) and a multiplicative
identity.  It seems unlikely that the term was chosen because of some
deeper algebraic understanding.  For example, despite listing 58
properties and theorems about matrices[1] the paper I cited does not
include that fact that multiplication distributes over addition.

[1] Including an unproved version of what came to be called the
Cayley-Hamilton theorem: that a square matrix satisfies its own
characteristic equation p(t) = det(tI - A).

Hi, Ben:

Cayley's 1858 paper about "matrix multiplication" occurs
after his 1854 attempt to define "group" in an abstract
way:

[The abstract group concept]
http://www.gap-system.org/~history/HistTopics/Abstract_groups.html

So group theory forms a more plausible "background"
for Cayley's choice of words than ring theory.

regards, chip

A

#### Ashton K

In sci.math BillyGates said:
I'm curious as to why it matters. Can you give me a good argument why
knowing the true etymology has any real significance in mathematics? We can
propose several likely reasons that are all logically coherent. Is it really
important to know which one is historically true? We can imagine that any of
the logical reasons could have be used in any alternate history and being
"wrong" here has no negative impact on our intellectual evolution?

Curiosity, really. It's what we humans do, be curious.

--Ashton

B

#### BillyGates

Ashton said:
Curiosity, really. It's what we humans do, be curious.

Cat's have curiosity too....

S

#### Seebs

Curiosity, really. It's what we humans do, be curious.

And in fact, it does rather affect how we learn -- curiousity and wanting to
fit things together and understand how they work is a very useful trait
in both programming and mathematics.

-s

H

#### Herman Jurjus

Ben said:
We can start observing that there is the Â»naturalÂ«
So, therefore, this is called Â»additionÂ«.

I assume that the algebraic laws for matrix
multiplication then relate it to matrix addition
in the way multiplication is related to addition
in a ring (which is the most general algebraic
structure with a multiplication and commutative
addition that I know). So, therefore, it then would
be Â»the multiplication in the ring of matricesÂ«.

Indeed, I now have confirmed via Wikipedia that
the matrices over a ring are a ring themselves.

s/matrices/NxN matrices/ (i.e. what you say is only true of square
matrices of a particular size).

Overall, the above sounds like a post-justification rather than a
probably explanation of the term. Ring theory started with
Dedekind in about 1870 and it was not until the early 20th century
that rings were unified by axiomatising the abstract structure
they all share.

I get the feeling that Cayley called his composition operation
"multiplication" because he observed that, for square matrices, there
is both a zero (additive and multiplicative) and a multiplicative
identity. It seems unlikely that the term was chosen because of some
deeper algebraic understanding. For example, despite listing 58
properties and theorems about matrices[1] the paper I cited does not
include that fact that multiplication distributes over addition.

Does your Cayley book mention complex numbers or their 'standard'
representation as 2x2 real matrices? Because that may also explain the
term: as a generalization of complex number multiplication to general
matrices.

B

#### Ben Bacarisse

Herman Jurjus said:
Ben said:
I get the feeling that Cayley called his composition operation
"multiplication" because he observed that, for square matrices, there
is both a zero (additive and multiplicative) and a multiplicative
identity. It seems unlikely that the term was chosen because of some
deeper algebraic understanding. For example, despite listing 58
properties and theorems about matrices[1] the paper I cited does not
include that fact that multiplication distributes over addition.

Does your Cayley book mention complex numbers or their 'standard'
representation as 2x2 real matrices? Because that may also explain the
term: as a generalization of complex number multiplication to general
matrices.

Not that I can see but the book is in two volumes and collects
together a very large number of papers most of which I have not even
scanned.

There are two papers on groups (not quite the groups we know today,
but very, very close) and he notes that one group is analogous (no
"isomorphic" yet) to the quaternions.

It seems improbable (to answer another poster here) that he did not
make the connection to a group so we can probably assume that he did.
The explanation as to why this is not mentioned in the matrix paper
is probably that infinite groups were not, at the time, as interesting
as differently structured finite groups. That's certainly the focus
of Cayley's group paper in the collection I've seen.

http://www.archive.org/details/collmathpapers01caylrich
http://www.archive.org/details/collmathpapers02caylrich