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Math Help - Solve for a matrix

  1. #1
    Junior Member
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    Solve for a matrix

    Assuming that all matrices are nxn and invertible, solve for D in the equation.

    ABC^TDBA^T = AB^T

    How can I do it?
    Please advise me the direction to solve this.

    Thank you.
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  2. #2
    Master Of Puppets
    pickslides's Avatar
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    I am a bit rusty on these ones but i'll give it a shot.

    We need to do some left multiplying on both sides

     ABC^TDBA^T = AB^T

     A^{-1}ABC^TDBA^T = A^{-1}AB^T

    gives

     IBC^TDBA^T = IB^T

     BC^TDBA^T = B^T

    try the same thing with B

     B^{-1}BC^TDBA^T = B^{-1}B^T

    gives

     IC^TDBA^T = B^{-1}B^T

     C^TDBA^T = B^{-1}B^T

    this is where it gets tricky I would try

     C^{T^{-1}}C^TDBA^T = C^{T^{-1}}B^{-1}B^T

    which gives

     IDBA^T = C^{T^{-1}}B^{-1}B^T

     DBA^T = C^{T^{-1}}B^{-1}B^T

    Now we need to do some right multiplying

     DBA^TA^{T^{-1}} = C^{T^{-1}}B^{-1}B^TA^{T^{-1}}

     DBI = C^{T^{-1}}B^{-1}B^TA^{T^{-1}}

     DB = C^{T^{-1}}B^{-1}B^TA^{T^{-1}}

    finally

     DBB^{-1} = C^{T^{-1}}B^{-1}B^TA^{T^{-1}}B^{-1}

     DI = C^{T^{-1}}B^{-1}B^TA^{T^{-1}}B^{-1}

     D = C^{T^{-1}}B^{-1}B^TA^{T^{-1}}B^{-1}

    I wouldn't take this as bible!



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