  Let $\langle m,n,p \rangle$ be the matrix multiplication tensor. The solution
set of Brent equations corresponds to the tensor decompositions of $\langle
m,n,p \rangle$. We study the local dimensions of solutions of the Brent
equations over the field of complex numbers. The rank of Jacobian matrix of
Brent equations provides an upper bound of the local dimension, which is
well-known. We calculate the ranks for some typical known solutions, which are
provided in the databases \cite{Faw22+} and \cite{Heule19}. We show that the
automorphism group of the natural algorithm computing $\langle m,n,p \rangle$
is $(\mathcal{P}_m\times \mathcal{P}_n\times \mathcal{P}_p)\rtimes Q(m,n,p)$,
where $\mathcal{P}_m$, $\mathcal{P}_n$ and $\mathcal{P}_p$ are groups of
generalised permutation matrices, $Q(m,n,p)$ is a subgroup of $S_3$ depending
on $m$, $n$ and $p$. For other algorithms computing $\langle m,n,p \rangle$,
some conditions are given, which imply the corresponding automorphism groups
are isomorphic to subgroups of $(\mathcal{P}_m\times \mathcal{P}_n\times
\mathcal{P}_p)\rtimes Q(m,n,p)$. So under these conditions,
$m^2+n^2+p^2-m-n-p-3$ is a lower bound for the local dimensions of solutions of
Brent equations. Moreover, the gap between the lower and upper bounds is
discussed.
