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Suppose $A$ is a positive definite $n \times n$ matrix over complex numbers. Then we know that $<Av,v>$ has minimum value $0$, when $v=0$.

Now suppose $v$ is constrained so that the first $k$ coordinates are fixed as some $k$-vector $c$ while the remaining $n-k$ coordinates are free.

What is the minimum of $<Av,v>$ in this case (in terms of the $k$-vector $c$ and appropriately chosen blocks of the matrix $A$?

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minimize to the most negtaive number ? –  Belgi Sep 5 '12 at 7:40
    
@Belgi: $<Av,v> > 0$ for any non-zero vector $v$ since $A$ is positive definite. When $v$ is constrained as above, the minimum can only get bigger. –  BharatRam Sep 5 '12 at 7:47
    
Maybe try to diagonalize $A$ using an orthonormal matrix, maybe since you can restrict the diagonal matrix into the last cooridinates it will be helpfull (although I can't yet see what happens to the inner product when I change $A$ to $P^*AP$) –  Belgi Sep 5 '12 at 7:51
    
That ends up distorting the vector, and doesnt immediately look promising. I did try it though, but I would like the answer in terms of blocks of $A$ and the vector $c$. –  BharatRam Sep 5 '12 at 7:59

1 Answer 1

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Write $A$ as a block matrix $\pmatrix{B & C\cr C^* & D\cr}$ and correspondingly $v = \pmatrix{c\cr w\cr}$. Then $$\langle A v, v \rangle = \langle B c, c \rangle + 2 \text{Re} \langle C^* c , w \rangle + \langle D w, w \rangle$$ Now $\langle D w, w \rangle = \langle D^{1/2} w, D^{1/2} w \rangle$. If we write $w = D^{-1/2} u$ then $$\langle A v, v \rangle = \langle Bc, c\rangle + 2 \text{Re} \langle D^{-1/2} C^* c, u \rangle + \langle u, u \rangle$$ Now given $<u, u>$, by Cauchy-Schwarz the minimum of $2 \text{Re} \langle D^{-1/2} C^* c, u\rangle$ with respect to $u$ is when $u$ is a negative multiple of $D^{-1/2} C^* c$. For $u = -t D^{-1/2} C^* c$ with $t > 0$ we have $\langle D^{-1/2} C^* c, u \rangle = - t \|D^{-1/2} C^* c \|^2$ and thus $$ \langle A v, v \rangle = \langle Bc, c\rangle + (t^2 - 2 t) \|D^{-1/2} C^* c\|^2 $$ To minimize $t^2 - 2 t$ we take $t = 1$, so that the minimum value of $\langle A v, v \rangle$ is $$\langle B c, c \rangle - \|D^{-1/2} C^* c \|^2 = \langle (B - C D^{-1} C^*) c, c \rangle $$

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