This applies only to a square matrix, it means swaping the rows and columns.

In other words element m_{ij} becomes element m_{ji}

m_{00} |
m_{01} --> m_{10} |
m_{02} --> m_{20} |

m_{10} --> m_{01} |
m_{11} |
m_{12} --> m_{21} |

m_{20} --> m_{02} |
m_{21} --> m_{12} |
m_{22} |

The transpose of matrix M is normally denoted by M^{t}.

## Conjugate transpose

U^{*} = the conjugate transpose of U.

The conjugate transpose is also known as adjoint matrix, adjugate matrix, hermitian adjoint or hermitian transpose.

This only applies to a matrix whose elements are complex numbers, quaternions or octonions.

The conjugate matrix is the complex conjugate of each element.

The conjugate transpose is the conjugate matrix and the transpose of the matrix applied together.