Given a $k\times n$ integer primitive matrix $A$ (i.e., a matrix can be extended to an $n\times n$ unimodular matrix over the integers) with size of entries bounded by $\lambda$, we study the probability that the $m\times n$ matrix extended from $A$ by choosing other $m-k$ vectors uniformly at random from $\{0, 1, \ldots, \lambda-1\}$ is still primitive. We present a complete and rigorous proof that the probability is at least a constant for the case of $m\le n-4$. Previously, only the limit case for $\lambda\rightarrow\infty$ with $k=0$ was analysed in Maze et al. (2011), known as the natural density. As an application, we prove that there exists a fast Las Vegas algorithm that completes a $k\times n$ primitive matrix $A$ to an $n\times n$ unimodular matrix within expected $\tilde{O}(n^{\omega}\log \|A\|)$ bit operations, where $\tilde{O}$ is big-$O$ but without log factors, $\omega$ is the exponent on the arithmetic operations of matrix multiplication and $\|A\|$ is the maximal absolute value of entries of $A$.

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关于生成原始矩阵的概率

给定一个$ k \ n的整数本原矩阵$ A $（即，一个矩阵可以在整数上扩展为一个$ n \ n的单模矩阵），且条目的大小由$ \ lambda $限制，我们研究了通过从$ \ {0，1，\ ldots，\ lambda-1 \} $中随机随机选择其他$ mk $向量，从$ A $扩展$ m \ timesn $矩阵的概率仍然是原始的。我们提供了一个完整而严格的证据，证明对于$ m \ le n-4 $的情况，概率至少是一个常数。以前，在Maze等人中只分析了$ \ lambda \ rightarrow \ infty $且$ k = 0 $的极限情况。（2011年），称为自然密度。作为一个应用程序，我们证明存在一种快速的拉斯维加斯算法，该算法可以在期望的$ \ tilde {O}（n ^ { \ omega} \ log \ | A \ |）$位操作，