$0\leq A \leq B$ means that $A$ and $B-A$ are positive as operators, i.e. $0\leq \langle Ax, x\rangle \leq \langle Bx, x\rangle$ for all $x\in \mathbb{C}^n$.
2026-03-30 14:59:03.1774882743
If we have $0\leq A \leq B$ for matrices $A$ and $B$, must $A$ and $B$ commute?
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Counterexample: $A=\pmatrix{1&0\\ 0&2}$ and $B=\pmatrix{3&1\\ 1&3}$.