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I would like to understand the following relation between unimodular matrices and its columns in some sense: if $x$ is a primitive vector (that is to say an integer column of $n$ rows whose entries are coprime), then it can be completed to an $n\times n$ unimodular matrix.

In the case of $2 \times 2$ matrices, I can see that it is equivalent to a Bezout relation, but is there a generalisation of this proof to show this property for all $n$?

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    $\begingroup$ This is equivalent to saying that the unimodular matrices act transitively on the integer "unit sphere" (the primitive vectors). If we know that any $x$ with GCD$(x)=1$ can be completed to a matrix $X$ with $|\det X|=1$, then given $x$ and $y$, we can define $A=YX^{-1}$ so that $AX=Y$ and, by selecting one column of this equation, $Ax=y$. Conversely, if we know that any primitive pair $x,y$ can be related by $Ax=y$ with $A$ unimodular, then we can take $y=[1,0,\cdots,0]^T$ so that $x=A^{-1}y$, which says that $x$ is the leftmost column of $A^{-1}$. $\endgroup$ – mr_e_man Jan 2 at 12:47
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This may not be what you want, but the completion can be done as follows.

Let $x=x_1$ be the given integer vector. We look for $2n-1$ integer vectors $x_2,\ldots,x_n,y_1,\ldots,y_n$ such that $$ Y^TX=\pmatrix{y_1^T\\ y_2^T\\ \vdots\\ y_n^T}\pmatrix{x_1&x_2&\cdots&x_n} =\pmatrix{1&\ast&\cdots&\ast\\ &1&\ddots&\vdots\\ &&\ddots&\ast\\ &&&1}. $$ Since both $X$ and $Y$ have integer determinants and $\det(Y)\det(X)=\det(Y^TX)=1$, if $X$ and $Y$ do exist, we must have $\det(X)=\pm1$.

We can construct the columns of $X$ and $Y$ by mathematical induction. In the base case, we pick an $y_1$ such that $y_1^Tx_1=1$. This is possible because the GCD of the entries of $x_1=x$ are coprime.

In the inductive step, suppose $1\le k<n$ and $x_1,\ldots,x_k,y_1,\ldots,y_k$ are such that $y_i^Tx_i=1$ for each $i\le k$ and $y_i^Tx_j=0$ whenever $j<i\le k$. Since the rank of $A=\pmatrix{x_1&\cdots&x_k}$ is smaller than $n$, the equation $y_{k+1}^TA=0$ has a nontrivial solution $y_{k+1}\in\mathbb Q^n$. By rationalising the denominators in its entries, $y_{k+1}$ can be chosen as an integer vector. Then by pulling out common factors in its entries, $y_{k+1}$ can be chosen to be primitive. Thus there exists an integer vector $x_{k+1}$ such that $y_{k+1}^Tx_{k+1}=1$ and our proof is complete.

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  • $\begingroup$ Can one further impose that the vectors $x_i$ for $i=2,\ldots,n$ are orthogonal to $x$ and still find a solution? $\endgroup$ – jj_p Nov 16 at 16:10
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    $\begingroup$ @jj_p No. E.g. if $\pmatrix{2&a\\ 3&b}$ is unimodular and it has orthogonal columns, then $2b-3a=\pm1$ and $2a+3b=0$. Hence $2(-2a/3)-3a=\pm1$, i.e. $a=\pm3/13\not\in\mathbb Z$. $\endgroup$ – user1551 Nov 16 at 18:07

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