= Matrix Transposition = For any matrix '''''A''''', the transposition ('''''A'''^T^'') is a flipped version. An alternative notation, found especially in matrix programming languages like [[Stata]], [[Julia]], and [[MATLAB]], is '''''A' '''''. <> ----- == Definition == Cell (''i'',''j'') of '''''A'''^T^'' is equal to cell (''j'',''i'') of '''''A'''''. {{{ julia> A = [1 2; 3 4] 2×2 Matrix{Int64}: 1 2 3 4 julia> A' 2×2 adjoint(::Matrix{Int64}) with eltype Int64: 1 3 2 4 }}} === Properties === The transpose of a product is the same as the reversed product of the transposed multiples. ''('''AB''')^T^ = '''B'''^T^ '''A'''^T^''. [[LinearAlgebra/MatrixInversion|Inversion]] and transposition can be done in any order: ''('''A'''^-1^)^T^ = ('''A'''^T^)^-1^''. For [[LinearAlgebra/Orthogonality#Matrices|orthogonal matrices]] (such as [[LinearAlgebra/SpecialMatrices#Permutation_Matrices|permutation matrices]]), the transpose is also the [[LinearAlgebra/MatrixInversion|inverse]]: '''''Q'''^T^ = '''Q'''^-1^''. And because the left and right inverses are the same for any square matrix, '''''QQ'''^T^ = '''Q'''^T^'''Q'''''. ---- == Symmetry == A [[LinearAlgebra/MatrixProperties#Symmetry|symmetric]] matrix is equal to its transpose: '''''A''' = '''A'''^T^''. Only square matrices (''n'' by ''n'') can be symmetric. However, multiplying a rectangular matrix '''''R''''' by its transpose '''''R'''^T^'' will always create a symmetric matrix. This can be proven with the above property: ''('''R'''^T^'''R''')^T^ = '''R'''^T^('''R'''^T^)^T^ = '''R'''^T^'''R'''''. ---- CategoryRicottone