Factorizing a matrix $(N\times N) \to (N-1 \times N) \times (N \times N-1)$

Question

I have a $N \times N $ matrix $H$, $rank(H)=N-1$.

I need to factorize $H$ into $O \times R$ such that $O$ has dimension $N \times N-1$

Since $H$ is not full rank this should always be possibile.

My questions are:

1) Is there some formula valid (at least when $N=2,3$) to speed up calculations?

2) Is this factorization unique?

Answer 1

Let $u$ be a basis of $\ker(H)$, $B_1=e_1,\cdots,e_{n-1},u$ be a basis of $K^n$ and $B_2=H(e_1),\cdots,H(e_{n-1}),v$ be another basis of $K^n$. With respect to these bases, we obtain $H=P\begin{pmatrix}I_{n-1}&0\\0&0\end{pmatrix}Q$ where $P=\begin{pmatrix}P_1&P_2\\P_3&P_4\end{pmatrix},Q=\begin{pmatrix}Q_1&Q_2\\Q_3&Q_4\end{pmatrix}\in GL_n(K)$.

Thus $H=OR=\begin{pmatrix}P_1\\P_3\end{pmatrix}\begin{pmatrix}Q_1&Q_2\end{pmatrix}$.