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Math Help - Positive Definiteness 2

  1. #1
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    Positive Definiteness 2

    Let A be any k x k matrix.
    Let X, Y be k x 1 vectors, such that XY'=YX'

    Let
     <br />
\[<br />
M =<br />
\left[ {\begin{array}{cc}<br />
 XX' & XY'  \\<br />
 YX' & YY'  \\<br />
 \end{array} } \right]<br />
\]<br /> <br />

    Let V= [A I-A]*M*[A' (I-A)']'

    Suppose that V-XX' is positive semidefinite, show that XX'=YX'

    Also, X and Y and linearly independent
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  2. #2
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    Quote Originally Posted by southprkfan1 View Post
    Let A be any k x k matrix.
    Let X, Y be k x 1 vectors, such that XY'=YX'

    Let
     <br />
\[<br />
M =<br />
\left[ {\begin{array}{cc}<br />
 XX' & XY'  \\<br />
 YX' & YY'  \\<br />
 \end{array} } \right]<br />
\]<br /> <br />

    Let V= [A I-A]*M*[A' (I-A)']'

    Suppose that V-XX' is positive semidefinite, show that XX'=YX'

    Also, X and Y and linearly independent
    I don't understand what's going on here. The given information seems to be self-contradictory. If XY' = YX' then X(Y'X) = Y(X'X). But Y'X and X'X are scalars, and X'X is nonzero. Therefore Y = \frac{Y'X}{X'X}X, which contradicts the information that X and Y are linearly independent.
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