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We give a simple construction of an orthogonal basis for the space of m by nmatrices with row and column sums equal to zero. This vector space correspondsto the affine space naturally associated with the Birkhoff polytope,contingency tables and Latin squares. We also provide orthogonal bases for thespaces underlying magic squares and Sudoku boards. Our construction combinesthe outer (i.e., tensor or dyadic) product on vectors with certain rooted,vector-labeled, binary trees. Our bases naturally respect the decomposition ofa vector space into centrosymmetric and skew-centrosymmetric pieces; the basescan be easily modified to respect the usual matrix symmetry and skew-symmetryas well.

Orthogonal bases for transportation polytopes applied to Latin squares, magic squares and Sudoku boards