On page 33 of B&V's convex optimization book, during the proof that any simplex can be represented as a polyhedron, they discuss a $n \times k$ matrix $B$ with full column rank and conclude that:
Therefore there exists a nonsingular matrix $A = (A_1, A_2) \in \mathbf{R}^{n \times n}$ such that $$ AB = \begin{bmatrix} A_1\\ A_2 \end{bmatrix} \;\; B = \begin{bmatrix} I\\ 0 \end{bmatrix} $$
This seems strange given the definition of $A$, and I can't find any reason from the text that B should have that structure, and from a later equation it appears that what they intended to say was: $$ A = \begin{bmatrix} A_1\\ A_2 \end{bmatrix} \;\; AB = \begin{bmatrix} I\\ 0 \end{bmatrix} $$ though this is not currently in the errata.
My questions are:
- Am I correct that this is indeed a mistake? If not, what am I misunderstanding?
- If it is, can $A$ be constructed by simply taking $A_1$ as any left inverse of $B$ and the rows of $A_2$ as any basis of the null space of $B^\intercal$? Would that guarantee that $A$ is nonsingular?