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= MatrixTransposition = = Transposition =

'''Transposition''' is the process of 'flipping' a matrix.

<<TableOfContents>>

-----
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== Introduction == == Description ==
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The transpose of a matrix is a flipped version. 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'''''.
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┌ ┐ ┌ ┐
│ 1 2│ │ 1 3│
│ 3 4│ -> │ 2 4│
└ ┘ └ ┘		 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
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The transpose of A is denoted A^T^.
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== Multiplication of Transposed Matrices == === Properties ===
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The transpose of a product is the same as the reversed product of the transposed multiples. (A B)^T^ = B^T^ A^T^. The transpose of a product is the same as the reversed product of the transposed multiples. ''('''AB''')^T^ = '''B'''^T^ '''A'''^T^''.
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[[LinearAlgebra/Invertibility|Inversion]] and transposition can be done in any order: ''('''A'''^-1^)^T^ = ('''A'''^T^)^-1^''.
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For [[LinearAlgebra/Orthogonality|orthogonal matrices]] (such as [[LinearAlgebra/SpecialMatrices#Permutation_Matrices|permutation matrices]]), the transpose and inverse are equivalent: '''''Q'''^T^ = '''Q'''^-1^''.
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== Inverses of Transposed Matrices ==

A simple proof based on the definition of inverse matrices and the above multiplicative property:

{{{
      -1 -1
   A A = I = A A

(leave the left side off for now)

      -1
   A A = I

       T
     -1 T
  A A = I

       T
     -1
  A A = I

   T
 -1 T
A A = I

(bring back the left side)

   T
 -1 T -1
A A = I = A A

(and it should now be clear that)

   T -1
 -1 T
A = A
}}}

Inverses and transposes can be done in any order.		 A [[LinearAlgebra/SpecialMatrices#Symmetric_Matrices|symmetric matrix]] is equal to its transpose: '''''A''' = '''A'''^T^''. Only square matrices can be symmetric.

Transposition

Transposition is the process of 'flipping' a matrix.

Contents

  1. Transposition
    1. Description
      1. Properties

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.

CategoryRicottone

LinearAlgebra/Transposition (last edited 2026-02-06 23:30:46 by DominicRicottone)