1. ## Positive Definiteness 2

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

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

$

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

2. Originally Posted by southprkfan1
Let A be any k x k matrix.
Let X, Y be k x 1 vectors, such that XY'=YX'

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

$

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
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.