Let $p$ be a parameter taking values in a compact set $P$. Let $M(p) = \begin{bmatrix} A(p) & D \\\\ B & C\end{bmatrix}$ be a block matrix. Further impose that $A(p)$ is an orthogonal matrix for every parameter value $p \in P$, and that $A(p)$ is Lipschitz inside $P$.
Questions:
- Is the pseudoinverse $p \mapsto M(p)^+$ continuous?
- If it is not continuous, is it lower semi-continuous, and does it have a continuous selecton?
- Does there exist a continuous (possibly non-unique) map from $p$ to the nullspace of $M(p)$?
Prior knowledge:
- The Pseudoinverse of a matrix is generically not continuous in its entries (https://en.wikipedia.org/wiki/Moore%E2%80%93Penrose_inverse)
- There exists an explicit formula for the pseudoinverse of a $2 \times 2$ block matrix Blockwise Moore-Penrose pseudoinverse?, https://www.sciencedirect.com/science/article/pii/0024379575901184
- Orthogonality of $A(p)$ implies that the upper block $[A(p), \ D]$ has full row rank.
- If $A$ is constant in $p$, then $M(p)$ is constant and the pseudoinverse is therefore continuous in $p$.