find a matrix such that $KX=0$

Question

Let $X$ be an $n \times p$ matrix. I would like to find a matrix $K$ such that $KX=0$. An obvious matrix is $K=I-X(X'X)^{-1}X'$.

I'm wondering if there are other matrices that satisfy $KX=0$?

Answer 1

Let $X^-$ be a generalised inverse of $X $. Then \begin{equation} (I-XX^- )X=0.\end{equation}

So if $X $ is not invertible, the answer will be that there are other $K $'s.

Answer 2

The condition $KX=0$ is equivalent to $\text{Im}\,X \subseteq \text{Ker}\,K$. Let $v_1,\ldots,v_k$ be an orthonormal basis of $\text{Im}\,X$ and $u_1,\ldots, u_l$ an orthonormal basis of $(\text{Im}\,X)^\perp$, so that $(v_1,\ldots,v_k,u_1,\ldots,u_l)$ is an ONB of $\mathbb R^n$

Then any $m\times n$ matrix of the form

$$K = \sum_{i=1}^l |\alpha_i\rangle\langle u_i| $$

where $\alpha_i\in\mathbb R^m$ suffices $KX=0$. Conversely, any matrix satisfying $KX=0$ is of the given form for some choice of the $\alpha_i$.