Let $A_1,\dots, A_k$ be non-zero $n \times n$ complex matrices, and let $1 \leq r \leq n$ be an integer. I want to know if there exists a polynomial time algorithm to decide if there exist $r \times n$ matrices $X,Y$ for which the following two properties hold:
- $X A_i Y^{T} \in \text{span}\{I_r\}$ for all $i=1,\dots,k$, and
- $X A_i Y^{T} =I_r$ for some $i =1,\dots, k$.
Here, $I_r$ is the $r \times r$ identity matrix.
Partial answer
In the special case $r=1$, such an $X$ and $Y$ always exist.
In the special case $r=n$, there is a polynomial time algorithm: If properties 1 and 2 are satisfied, then it must hold that $X$ and $Y$ are invertible $n \times n$ matrices. Using this, one can easily show that it is equivalent to ask whether the following linear system of equations has a solution ($n \times n$ matrix) $Z$:
- $Z A_i \in \text{span}\{I_n\}$ for all $i=1,\dots,k$, and
- $Z A_i =I_n$ for some $i=1,\dots,k$.
Motivation
This problem is motivated by the tensor subrank [Strassen]. It is unknown whether computing the tensor subrank is NP-hard, and this is a variant of that question.
