February 7, 2020

Roger Ngo

Let's take a matrix $\boldsymbol{A}$ representing some sort of column vector.

When we transpose a matrix, we want to find a new matrix such that the columns of matrix $\boldsymbol{M}$ become the rows of the matrix $\boldsymbol{M}^T$.

To find $\boldsymbol{A}^T$, we then need to take the columns in $\boldsymbol{A}$, and assign them as rows to $\boldsymbol{A}^T$.

For matrix $\boldsymbol{B}$:

We can then take each column, and transpose them individually to rows in $\boldsymbol{B}^T$.

This also works the other way around. You can also transpose the matrix by taking every row of the matrix, and assigning it as the column of the transposed matrix.

Suppose matrix $\boldsymbol{C} = \boldsymbol{B}^T$. Then $\boldsymbol{C}^T = (\boldsymbol{B}^T)^T$. Then, $\boldsymbol{C}^T = \boldsymbol{B}$.

Notice that we are not just limited to transposing square matrices.