Solve $X_2$ given $A(X_1;X_2)=B$

Question

Given that

1. $A$ is a $n \times n$ random orthogonal matrix, such that $A^TA=I$, $A^T1_n=1_n$, where $1_n$ is the vector of ones of size $n$.

2. $X = (X_1;X_2)$ is an $n \times p$ matrix with full column rank.

3. $X_1$ and $X_2$ are $\frac{n}{2} \times p$ matrices.

4. $A$, $X_2$ are unknown, while $X_1$ and $B$ are known.

Can we solve $X_2$ given $A(X_1;X_2) = B$?

Answer 1

Here is an example in the case $n=4$, $p=2$.

$$ X_1 = \pmatrix{1 & 0\cr 0 & 1\cr}, B = \pmatrix{1 & 0 \cr 0 & 1\cr 1 & 2\cr 3 & 4\cr} $$ Among the possibilities are

$$ A = \pmatrix{1 & 0 & 0 & 0\cr 0 & 1 & 0 & 0\cr 0 & 0 & 1 & 0\cr 0 & 0 & 0 & 1\cr},\ X_2 = \pmatrix{1 & 2\cr 3 & 4\cr} $$ and $$A = \pmatrix{1 & 0 & 0 & 0\cr 0 & 1 & 0 & 0\cr 0 & 0 & 0 & 1\cr 0 & 0 & 1 & 0\cr},\ X_2 = \pmatrix{3 & 4\cr 1 & 2\cr} $$

So no, you can't solve for $X_2$.