Source of term "multiplication" in matrix multiplication

Discussion in 'C Programming' started by William Hughes, Mar 13, 2010.

  1. 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
    William Hughes, Mar 13, 2010
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  2. William Hughes

    Chip Eastham Guest

    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

    But perhaps you are asking about the history of
    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
    Chip Eastham, Mar 13, 2010
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  3. William Hughes

    A N Niel Guest

    Wow. Instead of just guessing at the answer, looking up a reference?
    What a novel idea!
    A N Niel, Mar 13, 2010
  4. William Hughes

    BillyGates Guest

    A reference of someone guessing? Your right... that is novel!
    BillyGates, Mar 13, 2010
  5. 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

    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

    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.
    Ben Bacarisse, Mar 14, 2010
  6. William Hughes

    Stefan Ram Guest

    We can start observing that there is the »natural«
    way to »add« matrices, by adding the components.
    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.
    Stefan Ram, Mar 14, 2010
  7. 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).
    Ben Bacarisse, Mar 14, 2010
  8. William Hughes

    BillyGates Guest

    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?
    BillyGates, Mar 14, 2010
  9. William Hughes

    Chip Eastham Guest

    Hi, Ben:

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

    [The abstract group concept]

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

    regards, chip
    Chip Eastham, Mar 14, 2010
  10. William Hughes

    Ashton K Guest

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

    Ashton K, Mar 14, 2010
  11. William Hughes

    BillyGates Guest

    Cat's have curiosity too....
    BillyGates, Mar 14, 2010
  12. William Hughes

    Seebs Guest

    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.

    Seebs, Mar 14, 2010
  13. 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
    Herman Jurjus, Mar 15, 2010
  14. 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

    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.
    Ben Bacarisse, Mar 15, 2010
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