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Math Help - Orthogonality

  1. #1
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    Orthogonality

    How do I show ... if C and Q are nxn orthogonal matrices then so are CQ and (C^T)QC








    Find the closest line to the points (-1, 0); (0, 1); (1, 2); (2, 4).
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  2. #2
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    Quote Originally Posted by Noxide View Post
    How do I show ... if C and Q are nxn orthogonal matrices then so are CQ and (C^T)QC
    i'll do the first one since you can do the second in the same fashion.

    since C and Q were given as orthogonal matrices, then CC^t=C^tC=I and QQ^t=Q^tQ=I, so CQ(CQ)^t=CQQ^tC^t=I and (CQ)^tCQ=Q^tC^tCQ=I so CQ is orthogonal too.
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  3. #3
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    Quote Originally Posted by Noxide View Post
    How do I show ... if C and Q are nxn orthogonal matrices then so are CQ and (C^T)QC

    Find the closest line to the points (-1, 0); (0, 1); (1, 2); (2, 4).
    Second problem :

    It's curve-fit. To suppose line equation : y=ax+b,\vec{v}=\begin{bmatrix}a \\ b \end{bmatrix}, \begin{bmatrix}ax_1+b \\ax_2+b\\ax_3+b\\ax_4+b  \end{bmatrix}=\begin{bmatrix}x_1&1 \\x_2&1\\x_3&1\\x_4&1  \end{bmatrix}\begin{bmatrix}a\\b  \end{bmatrix}=A\vec{v}=\begin{bmatrix}0 \\1\\2\\4 \end{bmatrix}=\vec{y}

    Apply formulation  \vec{v}^*=(A^TA)^{-1}A^T\vec{y} =\frac{1}{10}\begin{bmatrix} 13 \\ 11  \end{bmatrix}

    10y=13x+11
    Last edited by math2009; November 2nd 2009 at 04:52 PM.
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