Transposing a matrix turns its rows into columns: the first row becomes the first column, the second row the second column, and so on. On the page it is a reflection of the block across its main diagonal.

A timetable with the days across the top can be reprinted with the days down the side. Nothing in it changes except the direction in which it is read.

The transpose of an $m\times n$ matrix $A$ is the $n\times m$ matrix $A^{T}$ whose row $i$ is column $i$ of $A$. A square matrix equal to its own transpose is symmetric, and one equal to the negative of its transpose is skew-symmetric. A vector written $(v_1,\ldots,v_n)$ in a sentence stands for a column, and its transpose is the same numbers written as a row.

• $(A^{T})^{T}=A$, $(A+B)^{T}=A^{T}+B^{T}$ and .

Compare entry $(i,j)$ of the two sides. On the left it is entry $(j,i)$ of $AB$: row $j$ of $$ against column of . On the right it is row of against column of , which is column of against row of — the same products, added in the same way.

Transposing moves entry $(j,i)$ into position $(i,j)$.

The row–column rule: row $j$ of $A$ against column $i$ of $B$.

Real numbers commute, and each factor is an entry of a transpose read with its indices swapped.

That sum is row $i$ of $B^{T}$ against column $j$ of $A^{T}$: the row–column rule for .

Every entry agrees, and the two sides have the same shape.

Write the first row of $A$ down the first column of $A^{T}$, the second row down the second column, and so on. An $m\times n$ matrix becomes $n\times m$.

Transposing a product transposes each factor and reverses their order. It is the rule for undoing two steps: the last one taken is the first one undone.

A square matrix is symmetric when it equals its own transpose. Entry by entry this says $a_{ij}=a_{ji}$: every entry equals the one in its mirror position across the main diagonal, and the diagonal itself can be anything.

If $A$ is invertible, transposing $AA^{-1}=I$ gives $(A^{-1})^{T}A^{T}=I^{T}=I$, and the other product works the same way. So is invertible, and its inverse is the transpose of .

• $(A^{T})_{ij}=a_{ji}$: rows become columns, and an matrix becomes .

Great work! You now know how to:

• transpose a matrix and locate the entries of a transpose;
• use $(AB)^{T}=B^{T}A^{T}$ and $$;