  The restricted invertibility theorem was originally introduced by Bourgain
and Tzafriri in $1987$ and has been considered as one of the most celebrated
theorems in geometry and analysis. In this note, we present weighted versions
of this theorem with slightly better estimates. Particularly, we show that for
any $A\in\mathbb{R}^{n\times m}$ and $k,r\in\mathbb{N}$ with $k\leq r\leq
\mbox{rank}(A)$, there exists a subset $\mathcal{S}$ of size $k$ such that
$\sigma_{\min}(A_{\mathcal{S}}W_{\mathcal{S}})^2\geq
\frac{(\sqrt{r}-\sqrt{k-1})^2}{\|W^{-1}\|_F^{2}}\cdot\frac{r}{\sum_{i=1}^{r}\sigma_{i}(A)^{-2}}$,
where $W=\mbox{diag}(w_1,\ldots,w_m)$ with $w_i$ being the weight of the $i$-th
column of $A$. Our constructions are algorithmic and employ the interlacing
families of polynomials developed by Marcus, Spielman, and Srivastava.
