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Math Help - Matrix expression question

  1. #1
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    Matrix expression question

    say Y=[y_1,y_2,...y_k], where y_j are column vectors. a=[a_1,a_2...a_k]^t is a column vector. Now I have the a polynomial \frac{1}{k}\sum_{j=1}^{k}a_j^2y_j^ty_j. Is it possible to express this polynomial in a form of Y and a?
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  2. #2
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    Try \frac{1}{k}a^t(y^ty)a
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  3. #3
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    You mean a^tY^tYa? No, that's not right.
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  4. #4
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    Reply to HallsofIvy:

    I'm not sure I agree that that will work.

    \displaystyle{\frac{1}{k}a^{t}(y^{t}y)a=\frac{1}{k  }(y^{t}y)(a^{t}a)=\frac{1}{k}\left(\sum_{j}y_{j}^{  t}y_{j}\right)\left(\sum_{l}a_{l}^{t}a_{l}\right).  }

    Is that necessarily equal to the desired expression? I'm not sure. I think you're going to have to use some sort of component-wise multiplication. Define the vector
    x\otimes y by

    (x\otimes y)_{j}=x_{j}y_{j}.

    Then you could write the desired expression as \displaystyle{\frac{1}{k}(a\otimes a)\cdot(y\otimes y).}
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  5. #5
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    I am not sure defining this vector can solve my problem. I wanted the polynomial in a matrix form because I need to do contained optimization with a^ta=1
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  6. #6
    A Plied Mathematician
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    Why don't you post your original question in full?
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  7. #7
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    Thank you for all your reply. I think I found a way to work around it. My problem is try to minimize the objective function \frac{1}{k}\sum_{j=1}^k(Xw-y_ja_j)^t(Xw-y_ja_j). w and a=[a_j] are variable vectors. I tried to solve w first then a, which gave me the question in this thread. Now I solve a_j first, then w has a closed form solution.
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  8. #8
    A Plied Mathematician
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    Well, if you've solved your problem, that's great! Have a good one.
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