Matrix Transposition

Transposition of a matrix means to pivot the matrix. If an mxnm x n matrix is given, the transposed matrix has the order nxmn x m. For example:

A=[1024619]→AT=[1041269]\mathbf{A} = \begin{bmatrix} 10 & 2 \\ 4 & 6 \\ 1 & 9 \end{bmatrix} \rightarrow \mathbf{A}^{T} = \begin{bmatrix} 10 & 4 & 1 \\ 2 & 6 & 9 \end{bmatrix}

The transposition of a matrix is notated using the TT superscript:

A→AT\mathbf{A} \rightarrow \mathbf{A}^{T}

Positional notation for a transpose is:

aijT=ajia_{ij}^{T} = a_{ji}

Properties of the transpose

  • (AT)T=A(\mathbf{A}^{T})^{T} = \mathbf{A}
  • (λA)T=λAT(\lambda\mathbf{A})^{T} = \lambda\mathbf{A}^{T}, where λ\lambda is a scalar
  • (A+B)T=AT+BT(\mathbf{A} + \mathbf{B})^{T} = \mathbf{A}^{T} + \mathbf{B}^{T}
  • (A+B+C)T=AT+BT+CT(\mathbf{A} + \mathbf{B} + \mathbf{C})^{T} = \mathbf{A}^{T} + \mathbf{B}^{T} + \mathbf{C}^{T}
  • (AB)T=BTAT(\mathbf{AB})^{T} = \mathbf{B}^{T}\mathbf{A}^{T}
  • (ABC)T=CTBTAT(\mathbf{ABC})^{T} = \mathbf{C}^{T}\mathbf{B}^{T}\mathbf{A}^{T}