Entropy and Hadamard matrices

We define the entropy of an orthogonal matrix Oⁱ_j. The entropy of the i^th row can have the maximum value ln n, which is attained when each element of the row is ±1/√n. This gives the bound, H{Oⁱ_j} ≤ n ln n. In general, the entropy of an orthogonal matrix cannot attain this bound because of the orthogonality constraint. In fact the bound is obtained only by the Hadamard matrices (rescaled by n^{- 1}2 /). Thus we have a new criterion for the Hadamard matrices (appropriately normalized): those orthogonal matrices which saturate the bound for entropy.