MathDB

7

Part of 2022 IMC

Problems(1)

Idempotent theoretical problem

Source: IMC 2022 Day 2 Problem 7

8/5/2022
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}.
linear algebraidempotencerankIMC 2022