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Math Help - Linear equations - Summation convention

  1. #1
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    Linear equations - Summation convention

    Hi, i want to show that zi=cij*xj, where cij=aik*bkj, if yi=bij*xj and zi=aij*yj. (i,j,k are subscripts).

    I have been struggling with this for a week. Any help is appreciated, thanks in advance.
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  2. #2
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    Re: Linear equations - Summation convention

    In matrix form, you want to show that y = Bx and z = Ay implies that z = (AB)x. From the assumption, it follows that z = A(Bx), so what you need to show is associativity of matrix multiplication: A(Bx) = (AB)x.

    Let A, B and C be matrices of compatible sizes. Then

    [AB]_{il}=\sum_k a_{ik}b_{kl} (1)

    and

    [BC]_{kj}=\sum_l b_{kl}c_{lj} (2)

    So we get

    \begin{align*}[(AB)C]_{ij} &= \sum_l\left(\sum_k a_{ik}b_{kl}\right)c_{lj} && \text{by (1)}\\ &=\sum_l\sum_k a_{ik}b_{kl}c_{lj}&& \text{by distributivity}\\ &=\sum_k\sum_l a_{ik}b_{kl}c_{lj}&& \text{by exchanging the order of summation, i.e., commutativity}\\ &=\sum_k a_{ik}\left(\sum_l b_{kl}c_{lj}\right)&& \text{by distributivity}\\ &=[A(BC)]_{ij}&& \text{by (2)} \end{align*}
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  3. #3
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    Re: Linear equations - Summation convention

    Hi, emakarov.

    Thank you for the answer.

    Is there any other way (more trivial) to prove that?
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  4. #4
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    Re: Linear equations - Summation convention

    Quote Originally Posted by FilipVz View Post
    Is there any other way (more trivial) to prove that?
    I am not sure. One can try writing the sums explicitly instead of using ∑. You can also try becoming more comfortable with ∑-notation and see what distributivity and exchanging limits means when written explicitly. Finally, you may search for explanations of the associativity of matrix multiplication; there must be good descriptions. (I am not sure if you realized that this problems boils down to associativity.)

    Feel free to post here what you tried or found.
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