# 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:  :$\displaystyle{ [\mathbf{A}^\mathrm{T}]_{ij} = [\mathbf{A}]_{ji} }$

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  :$\displaystyle{ \mathbf{A}^{\mathrm{T}} = \mathbf{A} . }$

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

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  :$\displaystyle{ \mathbf{A}^{\mathrm{T}} = \mathbf{A}^{*} . }$

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  :$\displaystyle{ \mathbf{A}^{\mathrm{T}} = -\mathbf{A}^{*} . }$

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