Transpose of a Matrix

Definition

Definition. The transpose of a matrix $A=[a_{ij}]$ of order $m\times n$ is the matrix obtained by interchanging its rows and columns: the $i$th row of $A$ becomes the $i$th column of the transpose, and vice versa. The transpose is denoted $A^T$ (or sometimes $A'$), and has order $n\times m$. Formally,

For example, if $A=\begin{bmatrix}1&2&3\\4&5&6\end{bmatrix}$ (order $2\times3$), then

(order $3\times2$) -- the first row $(1,2,3)$ of $A$ has become the first column of $A^T$, and the second row $(4,5,6)$ of $A$ has become the second column of $A^T$. Exercise: Transpose, Symmetric and Skew-Symmetric Matrices, Q1 practises this construction directly.

Properties of Transpose

For matrices $A,B$ (of orders for which the stated operations are defined) and scalar $k$, the transpose satisfies:

1. $(A^T)^T=A$ -- transposing twice returns the original matrix.
2. $(A+B)^T=A^T+B^T$ -- the transpose of a sum is the sum of the transposes (Example 7 verifies this numerically).
3. $(kA)^T=k\,A^T$ -- a scalar factor passes through transpose unchanged.
4. $(AB)^T=B^T A^T$ -- the reversal law: the transpose of a product equals the product of the transposes taken in reverse order. (This can be checked by comparing entries: the $(i,j)$ entry of $(AB)^T$ is $(AB)_{ji}=\sum_k a_{jk}b_{ki}$, while the $(i,j)$ entry of $B^TA^T$ is $\sum_k (B^T)_{ik}(A^T)_{kj}=\sum_k b_{ki}a_{jk}$ -- the same sum, term for term.)

Why Transpose Matters

Transpose is the tool that connects a matrix to two of the most important special classes studied in this chapter: symmetric and skew-symmetric matrices (Section 7), which are defined purely in terms of how a matrix compares to its own transpose, and it is also the operation used to check whether a candidate inverse genuinely satisfies the defining condition in some methods of computing an inverse (Section 9). …