# Horn’s inequality for singular values via exterior algebra

$$\DeclareMathOperator{\tr}{tr} \DeclareMathOperator{\sgn}{sgn} \DeclareMathOperator{\Id}{Id}$$Horn’s inequality states that for any two compact operators $$\sigma,\tau$$ on a Hilbert space $$E$$, $$\prod_{k=1}^n s_k(\sigma\tau) \le \prod_{k=1}^n s_k(\sigma)s_k(\tau)$$ where $$s_1(\tau),s_2(\tau),\dots$$ are the singular values of $$\tau$$ arranged in descending order. Alfred Horn’s original 1950 paper provides a short proof that relies on the following result: