= Transposition = '''Transposition''' is the process of 'flipping' a matrix. <> ----- == Description == A transposed matrix is commonly notated with a ''T'' superscript, as in '''''A'''^T^''. In many programming languages however, the notation '''''A' ''''' is preferred. 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/Invertibility|Inversion]] and transposition can be done in any order: ''('''A'''^-1^)^T^ = ('''A'''^T^)^-1^''. For [[LinearAlgebra/Orthogonality|orthogonal matrices]] (such as [[LinearAlgebra/SpecialMatrices#Permutation_Matrices|permutation matrices]]), the transpose and inverse are equivalent: '''''Q'''^T^ = '''Q'''^-1^''. A [[LinearAlgebra/SpecialMatrices#Symmetric_Matrices|symmetric matrix]] is equal to its transpose: '''''A''' = '''A'''^T^''. Only square matrices can be symmetric. For a matrix '''''A''''' of any size, '''''A'''^T^'''A''''' is always a symmetric, [[LinearAlgebra/Diagonalization|diagonalizable]], and [[LinearAlgebra/PositiveDefiniteness|positive definite]] matrix. Transposition does not change the [[LinearAlgebra/Determinant|determinant]] or the [[LinearAlgebra/Trace|trace]]: * ''|'''A'''| = |'''A'''^T^|'' * ''tr('''A''') = tr('''A'''^T^)'' ---- CategoryRicottone