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Thread: Where does matrix multipication come from?

  1. #1
    Jul 2008

    Where does matrix multipication come from?


    I am trying to get my head around the origin or matrix multiplication.

    Is a good description just: "Matrix Multiplication is just solving a system of linear equations. For example a nxm matrix 'A' (representing the coficients of n equations with m coefficients each) 'multiplies' by a mx1 vector 'V' (representing the variables), and the multiplication process is just solving for the variable; except in matrix multiplcation the the enries in V are actually constants and the anser vector holds the 'variables'. In the case when the vector V has more than one column this is just a a range of cases with different values for the enries in V. So, when V is a matrix, the answer to A x V, is a matrix with the number of columns equal to the number of cases set out as columns in the matrixs V." I just wrote this from the top of my head.

    I have been thinking about matrix multiplication like a mxnxk box where matrix A is mxn and matrix be is nxk. The front of the box is matrix A and the top is matrix B. Various points inside the box coincide with entries in A and B, and the values at these points are the product of the coresponding enrtries in a A and B. 'Surfaces' A and B of the box are joined along the side 'n' long. Summing along the n direction give values in a mxk matrix which is tyhe side of the box. This side is the answer of matirx multiplication.

    What is the role of determonants in matrix multiplication? I read in wikipedia that the determinant of a nxn matrix can be used to represent the volume inside a rhombus in n-space whose sides are made up of the entries in the matrix. Haven't thought about this enough yet.

    My question is can anyone give me some pointers?


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  2. #2
    Global Moderator

    Nov 2005
    New York City
    We can define matrix multiplication in an "obvious" way. Meaning just multiply the corresponding components. But this type of multiplication is not very interesting. The matrix multiplication that is would turns out being more useful. The simplest application of matrix multiplication is that we can regard a system of equations as simply $\displaystyle A\bold{x}=\bold{b}$. Another application is that linear transformations from $\displaystyle \mathbb{R}^m\mapsto \mathbb{R}^n$ can be again regarded as matrix multiplication. In fact, any linear transformation between two vector spaces in general can be expressed as a matrix. All of these connections follow by the way we define multiplication.
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  3. #3
    MHF Contributor Matt Westwood's Avatar
    Jul 2008
    Reading, UK
    Might be worth pointing out that matrix multiplication was defined the way it is for precisely the reason that it describes algebraically the geometry of linear transformations.
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