If $Q$ is unitary, and $I$ is the identity matrix, and $(I + Q)$ is invertible, do the matrices $(I - Q)$ and $(I + Q)^{-1}$ commute?
If so, is there a simple proof of this fact?
If $Q^* = Q^{-1}$, then is it necessarily true that $(I - Q)( I + Q)^{-1} = (I + Q)^{-1} (I - Q)$?
2
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linear-algebra
matrices
1 Answers
4
Yes, but this is independent of $Q$ being unitary. Suppose that $$(I+Q)(I-Q)=(I-Q)(I+Q).\tag{*}$$ If $I+Q$ is invertible, we can left-multiply and right-multiply the above equation by $(I+Q)^{-1}$ to obtain the desired equality, $$(I-Q)(I+Q)^{-1}=(I+Q)^{-1}(I-Q).$$ Therefore, it is sufficient to establish $(*)$ for an arbitrary matrix $Q$. Expanding $(*)$, we get $$I-Q+Q-Q^2=I+Q-Q-Q^2,$$ which is trivially true for any matrix $Q$.