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| Transposition does not change the [[LinearAlgebra/Determinant|determinant]] or the [[LinearAlgebra/Trace|trace]]: * ''|'''A'''| = |'''A'''^T^|'' * ''tr('''A''') = tr('''A'''^T^)'' |
Transposition
Transposition is the process of 'flipping' a matrix.
Contents
Description
A transposed matrix is commonly notated with a T superscript, as in AT. In many programming languages however, the notation A' is preferred.
Cell (i,j) of AT 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 = BT AT.
Inversion and transposition can be done in any order: (A-1)T = (AT)-1.
For orthogonal matrices (such as permutation matrices), the transpose and inverse are equivalent: QT = Q-1.
A symmetric matrix is equal to its transpose: A = AT. Only square matrices can be symmetric.
Transposition does not change the determinant or the trace:
|A| = |AT|
tr(A) = tr(AT)