Submatrices of Hadamard matrices: complementation results

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.