  We investigate conditions under which the identity matrix $I_n$ can be
continuously factorized through a continuous $N\times N$ matrix function $A$
with domain in $\mathbb{R}$. We study the relationship of the dimension $N$,
the diagonal entries of $A$, and the norm of $A$ to the dimension $n$ and the
norms of the matrices that witness the factorization of $I_n$ through $A$.
