# 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):****See:**Main Diagonal, Symmetric Matrix, Skew-Symmetric Matrix, Complex Number, Complex Conjugate, Hermitian Matrix, Conjugate Transpose, Skew-Hermitian Matrix, Orthogonal Matrix.

## 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},^{t}**A**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}

- reflect
- 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.

^{[1]}

- In linear algebra, the

- ↑ 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]

- A square matrix whose transpose is equal to itself is called a symmetric matrix; that is,