Let A be any k x k matrix.

Let X, Y be k x 1 vectors, such that XY'=YX'

Let

$\displaystyle

\[

M =

\left[ {\begin{array}{cc}

XX' & XY' \\

YX' & YY' \\

\end{array} } \right]

\]

$

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