Local and global bounds for the Shannon entropy of real orthogonal matrices have been established for various matrix orders. It has been shown that whenever a Hadamard matrix of order n exists, the entropy reaches its global maximum value of \(n \ln n\) . For other orders, where a Hadamard matrix does not exist, local maximal bounds are derived using orthogonal matrices associated with finite projective planes, biplanes, and triplanes. In the setting of complex inverse orthogonal matrices, explicit constructions are provided that attain this global maximum bound. Furthermore, the Rényi and Tsallis entropies for real orthogonal matrices are introduced, and both their local and global optimal bounds are determined for different values of n.
