  Given two symmetric and positive semidefinite square matrices $A, B$, is it
true that any matrix given as the product of $m$ copies of $A$ and $n$ copies
of $B$ in a particular sequence must be dominated in the spectral norm by the
ordered matrix product $A^m B^n$? For example, is $$ \| AABAABABB \| \leq \|
AAAAABBBB \|\ ? $$ Drury has characterized precisely which disordered words
have the property that an inequality of this type holds for all matrices $A,B$.
However, the $1$-parameter family of counterexamples Drury constructs for these
characterizations is comprised of $3 \times 3$ matrices, and thus as stated the
characterization applies only for $N \times N$ matrices with $N \geq 3$. In
contrast, we prove that for $2 \times 2$ matrices, the general rearrangement
inequality holds for all disordered words. We also show that for larger $N
\times N$ matrices, the general rearrangement inequality holds for all
disordered words, for most $A,B$ (in a sense of full measure) that are
sufficiently small perturbations of the identity.
