Let $A$ and $P$ be $n\times n$ matrices with $P$ a self-adjoint projection (i.e. $P=P^*=P^2$) and $\Vert A\Vert=1$ (where the norm here is the operator norm). Suppose that $A$ and $P$ almost commute, that is, the norm of their commutator $[A,P]$ is small. Is there a matrix $B$ near to $A$ in norm such that $[B,P]=0$? I am happy to assume that $A$ is self-adjoint if necessary.
By a famous theorem due to H. Lin (1993), there are commuting matrices $C$ and $D$ near to $A$ and $P$, but this doesn't immediately answer my question.
Yes. Note that $2P-I$ is a unitary matrix. Let $B=PAP+(I-P)A(I-P)$. Then $[B,P]=0$ and $$ \begin{aligned} \|A-B\| &=\|PA(I-P)+(I-P)AP\|\\ &=\left\|\big(PA(I-P)+(I-P)AP\big)\,(2P-I)\right\|\\ &=\left\|PA(I-P)(2P-I)+(I-P)AP(2P-I)\right\|\\ &=\left\|PA(P-I)+(I-P)AP\right\|\\ &=\|PAP-PA+AP-PAP\|\\ &=\|[A,P]\| \end{aligned} $$ is small.