Hadamard interchange of Y eigenstates ($H|\pm i⟩=e^{\pm i\pi /4}|\mp i⟩$)

For $|\pm i⟩=(|0⟩\pm i|1⟩)/\sqrt{2}$, direct matrix multiplication by the Hadamard gate gives coefficients $(1\pm i,1\mp i)/2$, proving the displayed identities. This exchanges the two Pauli eigenstates of the Pauli Y gate and agrees with $HYH=-Y$. The phases in the formula are global phases for each separate state; the relative phase inside each state is essential.