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Two submatrices A, D of a Hadamard matrix H are called complementary if, up to a permutation of rows and columns, H = [ A C B D ]. We find here an explicit formula for the polar decomposition of D. As an application, we show that under suitable smallness assumptions on the size of A, the complementary matrix D is an almost Hadamard sign pattern, i.e. its rescaled polar part is an almost Hadamard matrix. 2000 Mathematics Subject Classification. 15B34.doi:10.13001/1081-3810.1613 fatcat:wzqpx2thbjekji2nxlkvb6jbk4