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Let $T$ be a $d\times d$ matrix with real coefficients. Then $T$ determines aself-map of the $d$-dimensional torus ${\Bbb T}^d={\mathbb{R}}^d/{\Bbb Z}^d$.Let $ \{E_n \}_{n \in \mathbb{N}} $ be a sequence of subsets of ${\Bbb T}^d$and let $W(T,\{E_n \})$ be the set of points $\mathbf{x} \in {\Bbb T}^d$ suchthat $T^n(\mathbf{x})\in E_n $ for infinitely many $n\in {\mathbb{N}}$. For alarge class of subsets (namely, those satisfying the so called bounded property$ ({\boldsymbol{\rm B}}) $ which includes balls, rectangles, and hyperboloids)we show that the $d$-dimensional Lebesgue measure of the shrinking target set$W(T,\{E_n \})$ is zero (resp. one) if a natural volume sum converges (resp.diverges). In fact, we prove a quantitative form of this zero-one criteria thatdescribes the asymptotic behaviour of the counting function $R(x,N):= \# \big\{1\le n \le N : T^{n}(x) \in E_n \} $. The counting result makes use of ageneral quantitative statement that holds for a large class measure-preservingdynamical systems (namely, those satisfying the so called summable-mixingproperty). We next turn our attention to the Hausdorff dimension of $W(T,\{E_n\})$. In the case the subsets $E_n$ are balls, rectangles or hyperboloids weobtain precise formulae for the dimension. These shapes correspond,respectively, to the simultaneous, weighted and multiplicative theories ofclassical Diophantine approximation. The dimension results for ballsgeneralises those obtained in an earlier paper by Hill and the third-namedauthor for integer matrices to real matrices. In the final section, we discussvarious problems that stem from the results proved in the paper.

The Shrinking Target Problem for Matrix Transformations of Tori: revisiting the standard problem