  A pair $(A,\mathbf{b})$ of a real $m\times n$ matrix $A$ and
$\mathbf{b}\in\mathbb{R}^m$ is said to be $\textit{infinitely badly
approximable}$ if \[ \liminf_{\mathbf{q}\in\mathbb{Z}^n,
\|\mathbf{q}\|\to\infty}
\|\mathbf{q}\|^{\frac{n}{m}}\|A\mathbf{q}-\mathbf{b}\|_{\mathbb{Z}} =\infty, \]
where $\|\cdot\|_\mathbb{Z}$ denotes the distance from the nearest integer
vector. In this article, we introduce a novel concept of singularity for
$(A,\mathbf{b})$ and characterize the infinitely badly approximable property by
this singular property. As an application, we compute the Hausdorff dimension
of the infinitely badly approximable set. We also discuss dynamical
interpretations on the space of grids in $\mathbb{R}^{m+n}$.
