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Physically, one tends to think of non-Hermitian systems in terms of gain andloss: the decay or amplification of a mode is given by the imaginary part ofits energy. Here, we introduce an alternative avenue to the realm ofnon-Hermitian physics, which involves neither gain nor loss. Instead, complexeigenvalues emerge from the amplitudes and phase-differences of wavesbackscattered from the boundary of insulators. We show that for any strongtopological insulator in a Wigner-Dyson class, the reflected waves arecharacterized by a reflection matrix exhibiting the non-Hermitian skin effect.This leads to an unconventional Goos-H\"{a}nchen effect: due to non-Hermitiantopology, waves undergo a lateral shift upon reflection, even at normalincidence. Going beyond systems with gain and loss vastly expands the set ofexperimental platforms that can access non-Hermitian physics and showsignatures associated to non-Hermitian topology.

Non-Hermitian physics without gain or loss: the skin effect of reflected waves