Matrix Transpose

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A Matrix Transpose is a matrix that is the output of a matrix transpose operation.



References

2014

  • (Wikipedia, 2014) ⇒ http://en.wikipedia.org/wiki/Transpose Retrieved:2014-5-18.
    • In linear algebra, the transpose of a matrix A is another matrix AT (also written A′, Atr,tA or At) created by any one of the following equivalent actions:
      • reflect A over its main diagonal (which runs from top-left to bottom-right) to obtain AT
      • write the rows of A as the columns of AT
      • write the columns of A as the rows of AT
    • Formally, the i th row, j th column element of AT is the j th row, i th column element of A:  :[math]\displaystyle{ [\mathbf{A}^\mathrm{T}]_{ij} = [\mathbf{A}]_{ji} }[/math]

      If A is an matrix then AT is an matrix.

      The transpose of a matrix was introduced in 1858 by the British mathematician Arthur Cayley. [1]

  1. Arthur Cayley (1858) "A memoir on the theory of matrices," Philosophical Transactions of the Royal Society of London, 148 : 17-37. The transpose (or "transposition") is defined on page 31.


  • (Wikipedia, 2014) ⇒ http://en.wikipedia.org/wiki/transpose#Special_transpose_matrices Retrieved:2014-5-18.
    • A square matrix whose transpose is equal to itself is called a symmetric matrix; that is, A is symmetric if  :[math]\displaystyle{ \mathbf{A}^{\mathrm{T}} = \mathbf{A} . }[/math]

      A square matrix whose transpose is equal to its negative is called a skew-symmetric matrix; that is, A is skew-symmetric if  :[math]\displaystyle{ \mathbf{A}^{\mathrm{T}} = -\mathbf{A} . }[/math]

      A square complex matrix whose transpose is equal to the matrix with every entry replaced by its complex conjugate is called a Hermitian matrix (equivalent to the matrix being equal to its conjugate transpose); that is, A is Hermitian if  :[math]\displaystyle{ \mathbf{A}^{\mathrm{T}} = \mathbf{A}^{*} . }[/math]

      A square complex matrix whose transpose is equal to the negation of its complex conjugate is called a skew-Hermitian matrix; that is, A is skew-Hermitian if  :[math]\displaystyle{ \mathbf{A}^{\mathrm{T}} = -\mathbf{A}^{*} . }[/math]

      A square matrix whose transpose is equal to its inverse is called an orthogonal matrix; that is, A is orthogonal if  :[math]\displaystyle{ \mathbf{A}^{\mathrm{T}} = \mathbf{A}^{-1} . }[/math]