Assume that all matrices/vectors are real valued and
- $N < M$
- $B$ is a $N\times M$ matrix with rank $N$
- $Q$ is a $M \times (M-N)$ matrix. Figuring out properties/construction of $Q$ is the main goal.
- $u$ is column vector of length $M-N$
- $ \mathbf{0}_p$ is a column vector of $p$ zeros
A key property of $Q$ is that
First, for all $u\in \Bbb R^{M-N}$, $$ B Q u = \mathbf{0}_N $$
The matrix $P \equiv B Q$ is a $N \times (M-N)$ matrix.
Is there a named relationship between $Q$ and $B$ in the above (i.e. how is this related to the kernel, nullspace, etc.) That is, given a $Q$ how would I check if it fulfills this relationship for all $u$.
Furthermore, is there a named relationship between $Q$ and $B$ in the above (i.e., it is not the inverse, but is this related to annihilators?) If so, is there a way to construct a $Q$ for a given $B$, and is this unique (up to a scaling factor)?
Edit: previous mistakes in the sizes of matrices, and removed the second part of a question.
$Pu = 0$ for all $u\in\Bbb R^{M-N}$ means that $\ker P = \Bbb R^{M-N}$ i.e. $P=0$. Indeed, if $(e_1,...e_{M-N})$ is a basis of $\Bbb R^{M-N}$, then $\forall i$, the $i^{\text{th}}$ column of $P$ is $Pe_i = 0$, thus $P$ has only zero columns!
Thus, you want $Q$ such that $BQ = 0$ i.e. such that each column $C_i$ of $Q$ belongs to $\ker B$, i.e. $BC_i = 0,\forall i\in\{1,...,N\}$. Of course, a matrix $B$ satisfying this can't be unique (simply permute the columns or replace one of them by a linear combination of them or...).