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A new matrix product, called dimension-keeping semi-tensor product (DK-STP),is proposed. Under DK-STP, the set of $m\times n$ matrices becomes a semi-group$G({m\times n},\mathbb{F})$, and a ring, denoted by $R(m\times n,\mathbb{F})$.Moreover, the Lie bracket can also be defined, which turns the ring into a Liealgebra, called non-square (or STP) general linear algebra, denoted by$\mathrm{gl}(m\times n, \mathbb{F})$. Then the action of semi-group $G(m\timesn,\mathbb{F})$ on dimension-free Euclidian space, denoted by$\mathbb{R}^{\infty}$, is discussed. This action leads to discrete-time andcontinuous time S-systems. Their trajectories are calculated, and theirinvariant subspaces are revealed. As byproduct of this study, some importantconcepts for square matrices, such as eigenvalue, eigenvector, determinant,invertibility, etc., have been extended to non-square matrices. Particularly,it is surprising that the famous Cayley-Hamilton theory can also been extendedto non-square matrices. Finally, a Lie group, called the non-square (or STP)general Lie group and denoted by $\mathrm{GL}(m\times n,\mathbb{F})$, isconstructed, which has $\mathrm{GL}(m\times n,\mathbb{F})$ as its Lie algebra.Their relations with classical Lie group $\mathrm{GL}(m,\mathbb{F})$ and Liealgebra $\mathrm{gl}(m,\mathbb{F})$ are revealed.

From DK-STP to Non-square General Linear Algebras and General Linear Groups