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Math Help - posititve definite

  1. #1
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    posititve definite

    Can you, please help me with this. Even with the Hint I don't know whatto do
    Show that the following are equivalent
    (a) ( In U is posititve definite;
    U* Im)
    (b) Ip - UU* is positive definite;
    (c) Im - UU* is positive definite;
    Hint: Consider
    (v*, w*)( Ip U ) (v w)' = v*(Ip - UU*)v+?= w*(Im - UU*)w+?
    U* Im)
    Thank you!
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  2. #2
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    Quote Originally Posted by mivanova View Post
    Can you, please help me with this. Even with the Hint I don't know what to do
    Show that the following are equivalent
    (a) \begin{bmatrix}I_p&U\\U^*&I_m\end{bmatrix} is positive definite;
    (b) I_p - UU^* is positive definite;
    (c) I_m - U^*U is positive definite;
    Hint: Consider \begin{bmatrix}v^*&w^*\end{bmatrix} \begin{bmatrix}I_p&U\\U^*&I_m\end{bmatrix} \begin{bmatrix}v\\w\end{bmatrix} = v^*(I_p - UU^*)v+{}? = w^*(I_m - U^*U)w+{}?.
    First, note that I have made some corrections to the question. The most important one is that some of the UU*s should be U*U.

    Start by doing what the Hint suggests, and multiply the matrices to get the product

    \begin{bmatrix}v^*&w^*\end{bmatrix} \begin{bmatrix}I_p&U\\U^*&I_m\end{bmatrix} \begin{bmatrix}v\\w\end{bmatrix} = \begin{bmatrix}v^*&w^*\end{bmatrix} \begin{bmatrix}v+Uw\\U^*v+w\end{bmatrix} = v^*v+v^*Uw + w^*U^*v + w^*w.

    Now comes the clever part. The Hint suggests that you should write this in the form v^*(I_p - UU^*)v+{}?. But v^*(I_p - UU^*)v = v^*v - v^*UU^*v = v^*v - (U^*v)^*(U^*v) (because (U^*v)^* = v^*U). So we get

    \begin{aligned}\begin{bmatrix}v^*&w^*\end{bmatrix} \begin{bmatrix}I_p&U\\U^*&I_m\end{bmatrix} \begin{bmatrix}v\\w\end{bmatrix} &= v^*(I_p - UU^*)v + (U^*v)^*(U^*v) + (v^*U)w + w^*(U^*v) + w^*w \\ &= v^*(I_p - UU^*)v + (U^*v+w)^*(U^*v+w).\end{aligned}

    In the equation \begin{bmatrix}v^*&w^*\end{bmatrix} \begin{bmatrix}I_p&U\\U^*&I_m\end{bmatrix} \begin{bmatrix}v\\w\end{bmatrix} = v^*(I_p - UU^*)v + (U^*v+w)^*(U^*v+w), you can see that if I_p - UU^* is positive definite then the right side of the equation will be positive for all v and w. Therefore the left side is positive, which means that the matrix \begin{bmatrix}I_p&U\\U^*&I_m\end{bmatrix} is positive definite. On the other hand, if the matrix is positive definite then the left side of the equation is positive for all v and w, and in particular when w = -U^*v. That implies that v^*(I_p - UU^*)v is positive for all v, which tells us that I_p - UU^* is positive definite.

    Thus condition (a) is equivalent to condition (b). A similar calculation shows that (a) is equivalent to (c).
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