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Proving two simple matrix product properties

  1. Sep 4, 2016 #1
    1. The problem statement, all variables and given/known data
    Let ##A## be an n × p matrix and ##B## be an p × m matrix with the following column vector representation,

    [tex]
    B = \begin{bmatrix}
    b_1 , & b_2, & ... & ,b_m
    \end{bmatrix}
    [/tex]

    Prove that
    [tex]AB =
    \begin{bmatrix}
    Ab_1 , & Ab_2, & ... & , Ab_m
    \end{bmatrix}
    [/tex]

    If ##A## is represented with help of its row vectors, prove that

    [tex]
    AB =
    \begin{bmatrix}
    a^{T}_{1}\\
    \vdots\\
    a_{n}^{T}
    \end{bmatrix} B
    =
    \begin{bmatrix}
    a^{T}_{1} B\\
    \vdots\\
    a^{T}_{n} B
    \end{bmatrix}
    [/tex]

    2. Relevant equations

    The matrix product:
    If ##A## is an ##m\times p## matrix and ##B## is a ##p\times n## matrix, then
    \begin{equation}
    AB = C = (c_{ij})_{m \times n} = (\sum_{k=1}^{p} a_{ik}b_{kj})_{m \times n}
    \end{equation}

    3. The attempt at a solution

    For starters what does proving in this context mean? Should I simply write out the matrix

    [tex]
    C =
    \begin{bmatrix}
    \sum_{k=1}^{p} a_{1k} b_{k1} & \cdots & \sum_{k=1}^{p} a_{1k} b_{km}\\
    \vdots & \ddots & \vdots\\
    \sum_{k=1}^{p} a_{mk} b_{k1} & \cdots &\sum_{k=1}^{p} a_{mk} b_{km}
    \end{bmatrix}
    [/tex]
    and conclude that each column is essentially ##Ab_{l}## where ##1 < l < m##, since each element of the matrix is the dot (inner) product of a row in ##A## and a column in ##B##?
     
    Last edited: Sep 4, 2016
  2. jcsd
  3. Sep 4, 2016 #2
    Oops. This

    [tex]
    C =
    \begin{bmatrix}
    \sum_{k=1}^{p} a_{1k} b_{k1} & \cdots & \sum_{k=1}^{p} a_{1k} b_{km}\\
    \vdots & \ddots & \vdots\\
    \sum_{k=1}^{p} a_{mk} b_{k1} & \cdots &\sum_{k=1}^{p} a_{mk} b_{km}
    \end{bmatrix}
    [/tex]

    should be this
    [tex]
    C =
    \begin{bmatrix}
    \sum_{k=1}^{p} a_{1k} b_{k1} & \cdots & \sum_{k=1}^{p} a_{1k} b_{km}\\
    \vdots & \ddots & \vdots\\
    \sum_{k=1}^{p} a_{nk} b_{k1} & \cdots &\sum_{k=1}^{p} a_{mk} b_{km}
    \end{bmatrix}
    [/tex]
    since it's an ##n \times m## matrix, not an ##m \times m## matrix.
     
  4. Sep 8, 2016 #3

    EnumaElish

    User Avatar
    Science Advisor
    Homework Helper

    1. Calculate C = AB.
    2. Calculate D = [A b1 ... A bm]
    3. Show Dij = Cij for any i,j
     
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