  The rigidity of a matrix $A$ for target rank $r$ is the minimum number of
entries of $A$ that need to be changed in order to obtain a matrix of rank at
most $r$. Matrix rigidity was introduced by Valiant in 1977 as a tool to prove
circuit lower bounds for linear functions and since then this notion has also
found applications in other areas of complexity theory.
  Recently (arXiv 2021), Alman proved that for any field $\mathbb{F}$, $d\geq
2$ and arbitrary matrices $M_1, \ldots, M_n \in \mathbb{F}^{d\times d}$, one
can get a $d^n\times d^n$ matrix of rank $\le d^{(1-\gamma)n}$ over
$\mathbb{F}$ by changing only $d^{(1+\varepsilon) n}$ entries of the Kronecker
product $M = M_1\otimes M_2\otimes \ldots\otimes M_n$, where $1/\gamma$ is
roughly $2^d/\varepsilon^2$.
  In this note we improve this result in two directions. First, we do not
require the matrices $M_i$ to have equal size. Second, we reduce $1/\gamma$
from exponential in $d$ to roughly $d^{3/2}/\varepsilon^2$ (where $d$ is the
maximum size of the matrices), and to nearly linear (roughly $d/\varepsilon^2$)
for matrices $M_i$ of sizes within a constant factor of each other.
  For the case of matrices of equal size, our bound matches the bound given by
Dvir and Liu (\textit{Theory of Computing, 2020}) for the rigidity of
generalized Walsh--Hadamard matrices (Kronecker powers of DFT matrices), and
improves their bounds for DFT matrices of abelian groups that are direct
products of small groups.
