Transpose of a Matrix

The transposition of a matrix (or transpose of matrix) is one of the most basic matrix operations to perform. The transpose of a matrix consists of inverting the rows with the columns:

$$ \text{ If } M = \begin{bmatrix} a & c & e \\ b & d & f \end{bmatrix} \text{ Then } M^T = \begin{bmatrix} a & b \\ c & d \\ e & f \end{bmatrix} $$

The lines are read from left to right and are transposed from top to bottom.

Example: $$ M = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} \Rightarrow M^t = \begin{bmatrix} 1 & 3 \\ 2 & 4 \end{bmatrix} $$

The transposition of a matrix \( M \) is noted \( M^t \) or \( ^tM \). The transposition operation is then noted with an exponent T or t (uppercase or lowercase) prefixed or postfixed.

The transposition is valid on both square matrices and rectangular matrices. A transposed row vector is a column vector and vice versa.

Transposing twice a matrix returns it unchanged.

The double transposition is the name given to a cryptographic cipher.

The transpose of a column matrix is a line matrix of the same size and vice versa.

Example: The transpose from \( \begin{bmatrix} a \\ b \end{bmatrix} \) is \( \begin{bmatrix} a & b \end{bmatrix} \)

Example: The transpose from \( \begin{bmatrix} a & b \end{bmatrix} \) is \( \begin{bmatrix} a \\ b \end{bmatrix} \)