## 每週問題 October 7, 2013

Let $\mathbf{x},\mathbf{y}\in \mathbb{C}^n$. If $\Vert\mathbf{x}\Vert< 1$ and $\Vert\mathbf{y}\Vert< 1$, show that

$\displaystyle \begin{bmatrix} \frac{1}{1-\left\langle\mathbf{x},\mathbf{x}\right\rangle}&\frac{1}{1-\left\langle\mathbf{x},\mathbf{y}\right\rangle}\\[0.5em] \frac{1}{1-\left\langle\mathbf{y},\mathbf{x}\right\rangle}&\frac{1}{1-\left\langle\mathbf{y},\mathbf{y}\right\rangle} \end{bmatrix}$

is positive semidefinite.

$\displaystyle \vert \left\langle\mathbf{x},\mathbf{y}\right\rangle\vert\le\Vert\mathbf{x}\Vert\cdot\Vert\mathbf{y}\Vert$

$\displaystyle \frac{1}{1-r}=\sum_{k=0}^\infty r^k$

$\displaystyle \begin{bmatrix} \frac{1}{1-\left\langle\mathbf{x},\mathbf{x}\right\rangle}&\frac{1}{1-\left\langle\mathbf{x},\mathbf{y}\right\rangle}\\[0.5em] \frac{1}{1-\left\langle\mathbf{y},\mathbf{x}\right\rangle}&\frac{1}{1-\left\langle\mathbf{y},\mathbf{y}\right\rangle} \end{bmatrix}=\begin{bmatrix} \sum_{k=0}^\infty \left\langle\mathbf{x},\mathbf{x}\right\rangle^k&\sum_{k=0}^\infty \left\langle\mathbf{x},\mathbf{y}\right\rangle^k\\[0.5em] \sum_{k=0}^\infty \left\langle\mathbf{y},\mathbf{x}\right\rangle^k&\sum_{k=0}^\infty \left\langle\mathbf{y},\mathbf{y}\right\rangle^k \end{bmatrix}=\sum_{k=0}^\infty\begin{bmatrix} \left\langle\mathbf{x},\mathbf{x}\right\rangle^k&\left\langle\mathbf{x},\mathbf{y}\right\rangle^k\\[0.5em] \left\langle\mathbf{y},\mathbf{x}\right\rangle^k&\left\langle\mathbf{y},\mathbf{y}\right\rangle^k \end{bmatrix}$

$\displaystyle \mathbf{z}^\ast (A+B)\mathbf{z}=\mathbf{z}^\ast A\mathbf{z}+\mathbf{z}^\ast B\mathbf{z}\ge 0$