Transpose, Conjugate, Conjugate Transpose

Sometimes we want to take the transpose of a matrix. 

What this means is, row 1 of the matrix becomes column 1, row 2 becomes column 2, and so on (and vice-versa). For example, consider the matrix:

$$A = \begin{bmatrix} 1 & 3 \\ 7 & 9 \\ 2 & 8 \end{bmatrix}.$$

Then the transpose of this matrix, denoted $A^T$, is:

$$A^T = \begin{bmatrix} 1 & 7 & 2 \\ 3 & 9 & 8 \end{bmatrix}.$$

Note that $A$ had dimensions $3 \times 2$, and $A^T$ has dimensions $2 \times 3$.

This is true in general, i.e., if $A$ is an $m \times n$ matrix then $A^T$ is an $n \times m$ matrix, which makes sense since taking the transpose amounts to swapping the rows and columns of the original matrix.

Also note that taking the transpose of a column vector gives a row vector and vice-versa, i.e., for the column vector $\vec{v} = \begin{bmatrix} 1 \\ 0 \\ -1 \\ -2 \end{bmatrix}$ we have $\vec{v}^T = \begin{bmatrix} 1 & 0 & -1 & -2 \end{bmatrix}$.

It's also worth noting that taking the transpose twice is equivalent to doing nothing. That is, $(A^T)^T = A$.

Indeed:

$$\begin{align*} (A^T)^T &= \begin{bmatrix} 1 & 7 & 2 \\ 3 & 9 & 8 \end{bmatrix}^T \\ &= \begin{bmatrix} 1 & 3 \\ 7 & 9 \\ 2 & 8 \end{bmatrix} \\ &= A \end{align*}$$

and:

$$...$$