calculate orthogonal matrix

Question

Given matrix $A(m\times n)$

find matrix $B(n\times m)$ that fulfill the equation

$A\,B=0\,(m\times m)$ mean orthogonal

m less then n

Answer 1

Use $A^+$ (the pseudoinverse of $A$) to construct the ($n\times n$) nullspace projector $$P = I_n - A^+A$$ The key property of this projector is $$AP = (A-AA^+A) = (A-A) = 0$$ Now any ($n\times m$) matrix $C$ can be used to construct the required matrix $$B = PC$$ NB:  For $m\ge n,\,$ $A^+A=I\,$ and therefore $P=0\,$ and $B=0$.