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Idempotent theoretical problem

Source: IMC 2022 Day 2 Problem 7

August 5, 2022
linear algebraidempotencerankIMC 2022

Problem Statement

Let A1,…,AkA_1, \ldots, A_k be n×nn\times n idempotent complex matrices such that AiAj=−AiAjA_iA_j = -A_iA_j for all 1≤i<j≤k1 \leq i < j \leq k. Prove that at least one of the matrices has rank not exceeding nk\frac{n}{k}.
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