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It is known that semi-magic square matrices form a 2-graded algebra orsuperalgebra with the even and odd subspaces under centre-point reflectionsymmetry as the two components. We show that other symmetries which have beenstudied for square matrices give rise to similar superalgebra structures,pointing to novel symmetry types in their complementary parts. In particular,this provides a unifying framework for the composite `most perfect square'symmetry and the related class of `reversible squares'; moreover, thesemi-magic square algebra is identified as part of a 2-gradation of the generalsquare matrix algebra. We derive explicit representation formulae for matricesof all symmetry types considered, which can be used to construct all suchmatrices.

On Superalgebras of Matrices with Symmetry Properties