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Math Help - Help with a problem

  1. #1
    Senior Member Pinkk's Avatar
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    Help with a problem

    Assuming that all matrices are n\,\,x\,\,n and invertible, solve for D:

    ABC^{T}DBA^{T}C=AB^{T}

    I realized that I would someone have to reduce everything to the right of D to B^{T}I_{n} and everything to the left would be AI_{n}

    So for the "right side", I started with C^{-1}A^{-T}B^{-1} =(BA^{T}C)^{-1} since C^{-1}A^{-T}B^{-1}BA^{T}C=I_{n} and therefore:

    B^{T}(BA^{T}C)^{-1} would result in B^{T}I_{n}=B^{T}

    For the "left size", I started with C^{-T}B^{-1}=(BC^{T})^{-1} since ABC^{T}C^{-T}B^{-1}=AI_{n}=A

    Multiplying the "left" with the "right" to satisfy the whole equation, I ended up with:

    D=(BC^{T})^{-1}B^{T}(BA^{T}C)^{-1}

    Is this correct, or at the very least is my though process correct and if so, what possible errors are there? Thank you.
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  2. #2
    Lord of certain Rings
    Isomorphism's Avatar
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    Quote Originally Posted by Pinkk View Post
    Assuming that all matrices are n\,\,x\,\,n and invertible, solve for D:

    ABC^{T}DBA^{T}C=AB^{T}

    I realized that I would someone have to reduce everything to the right of D to B^{T}I_{n} and everything to the left would be AI_{n}

    So for the "right side", I started with C^{-1}A^{-T}B^{-1} =(BA^{T}C)^{-1} since C^{-1}A^{-T}B^{-1}BA^{T}C=I_{n} and therefore:

    B^{T}(BA^{T}C)^{-1} would result in B^{T}I_{n}=B^{T}

    For the "left size", I started with C^{-T}B^{-1}=(BC^{T})^{-1} since ABC^{T}C^{-T}B^{-1}=AI_{n}=A

    Multiplying the "left" with the "right" to satisfy the whole equation, I ended up with:

    D=(BC^{T})^{-1}B^{T}(BA^{T}C)^{-1}

    Is this correct, or at the very least is my though process correct and if so, what possible errors are there? Thank you.
    Yes it is

    You can check it yourself, By substituting your D in the first equation and seeing if it satisfies
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  3. #3
    Senior Member Pinkk's Avatar
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    Uptown Manhattan, NY, USA
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    Thank you. I checked it and it did work, but I wanted to make sure I was conveying the theorems of matrix invertibility, etc correctly. I'm having a bit harder time grasping linear algebra than I did grasping Calc 1,2, and 3.
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