Matrix Transpose

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

• Context:
  • It can be represented with [math]\displaystyle{ \mathbf{A}^{\mathrm{T}} }[/math].
• Example(s):
  • for a symmetric matrix [math]\displaystyle{ \mathbf{A}^{\mathrm{T}} = \mathbf{A} . }[/math]
• Counter-Example(s):
  • Matrix Inverse.
• See: Main Diagonal, Symmetric Matrix, Skew-Symmetric Matrix, Complex Number, Complex Conjugate, Hermitian Matrix, Conjugate Transpose, Skew-Hermitian Matrix, Orthogonal Matrix.

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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 A^T (also written A′, A^tr,^tA or A^t) 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 A^T
    • write the rows of A as the columns of A^T
    • write the columns of A as the rows of A^T
  • Formally, the i th row, j th column element of A^T 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 A^T is an matrix.

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

• (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]

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