Submatrices of Hadamard matrices: complementation results

Abstract

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.

The quantum algebra of partial Hadamard matrices

“…For more information on minors of Hadamard matrices we refer to the articles [10], [11], see also [4], where square submatrices of Hadamard matrices are considered.…”

“…On the other hand, checking that some projections are pairwise orthogonal is the same as checking that their sum is a projection, and this gives (2) ⇐⇒ (4) and (3) ⇐⇒ (4), because the projection in question is 1 − P N N . Finally, we have (4) =⇒ (1), because we can complete P with the projections in (2,3,4).…”

Analytic aspects of the circulant Hadamard conjecture

“…The study of these critical points falls into the "almost Hadamard matrix" framework from [5], [8], [10]. For some recent advances here, see [9].…”

“…- (5,6), (6,6), (6,9), (8,6). Here the application of the Turyn obstruction is a more complicated task, and we obtained the results by using a computer.…”

Embedding and Extension Properties of Hadamard Matrices Revisited

“…An Hadamard matrix of order n, denoted by Hn, is an n × n matrix with elements + or − and mutually orthogonal rows and columns, i.e., Hn H n = H n Hn = n In (1) where H n denotes the transpose of Hn and In is the identity matrix of order n. Also, a Hadamard matrix is said to be normalized if it has its rst row and column all 1's. Hadamard himself showed that the matrices of this kind have the maximal determinant | det Hn| = n n/ (2) and he observed that such matrices could exist only if n was 1, 2 or a multiple of 4 [8].…”