2.5: The Transpose

Another important operation on matrices is that of taking the transpose. For a matrix $A$, we denote the transpose of $A$ by $A^T$. Before formally defining the transpose, we explore this operation on the following matrix.

$$\left[ \begin{array}{cc} 1 & 4 \\ 3 & 1 \\ 2 & 6 \end{array} \right] ^{T}= \ \ \left[ \begin{array}{ccc} 1 & 3 & 2 \\ 4 & 1 & 6 \end{array} \right] \nonumber$$

What happened? The first column became the first row and the second column became the second row. Thus the $3\times 2$ matrix became a $2\times 3$ matrix. The number $4$ was in the first row and the second column and it ended up in the second row and first column.

The definition of the transpose is as follows.

Definition $\PageIndex{1}$: The Transpose of a Matrix

Let $A$ be an $m\times n$ matrix. Then $A^{T}$, the transpose of $A$, denotes the $n\times m$ matrix given by

$$A^{T} = \left[ a _{ij}\right] ^{T}= \left[ a_{ji} \right]\nonumber$$

The $\left( i, j \right)$-entry of $A$ becomes the $\left( j,i \right)$-entry of $A^T$.

Consider the following example.

Example $\PageIndex{1}$: The Transpose of a Matrix

Calculate $A^T$ for the following matrix

$$A = \left[ \begin{array}{rrr} 1 & 2 & -6 \\ 3 & 5 & 4 \end{array} \right] \nonumber$$

Solution

By Definition $\PageIndex{1}$, we know that for $A = \left[ a_{ij} \right]$, $A^T = \left[ a_{ji} \right]$. In other words, we switch the row and column location of each entry. The $\left( 1, 2 \right)$-entry becomes the $\left( 2,1 \right)$-entry.

Thus,

$$A^T = \left[ \begin{array}{rr} 1 & 3 \\ 2 & 5 \\ -6 & 4 \end{array} \right] \nonumber$$

Notice that $A$ is a $2 \times 3$ matrix, while $A^T$ is a $3 \times 2$ matrix.

The transpose of a matrix has the following important properties.

Lemma $\PageIndex{1}$: Properties of the Transpose of a Matrix

Let $A$ be an $m\times n$ matrix, $B$ an $n\times p$ matrix, and $r$ and $s$ scalars. Then

1. $$\left(A^{T}\right)^{T} = A\nonumber$$
2. $$\left( AB\right) ^{T}=B^{T}A^{T} \nonumber$$
3. $$\left( rA+ sB\right) ^{T}=rA^{T}+ sB^{T} \nonumber$$

Proof

First we prove 2. From Definition $\PageIndex{1}$,

$$\begin{aligned} \left(AB\right)^{T} &= \left[ (AB) _{ij} \right] ^{T}=\left[ (AB)_{ji} \right]=\sum_{k}a_{jk}b_{ki}= \sum_{k}b_{ki}a_{jk} \\[4pt] &= \sum_{k}\left[ b_{ik}\right]^{T}\left[ a_{kj}\right]^{T}=\left[ b_{ij}\right] ^{T} \left[ a_{ij}\right]^{T} = B^{T}A^{T} \end{aligned}$$

The proof of Formula 3 is left as an exercise.

The transpose of a matrix is related to other important topics. Consider the following definition.

Definition $\PageIndex{2}$: Symmetric and Skew Symmetric Matrices

An $n\times n$ matrix $A$ is said to be symmetric if $A=A^{T}.$ It is said to be skew symmetric if $A=-A^{T}.$

We will explore these definitions in the following examples.

Example $\PageIndex{2}$: Symmetric Matrices

Let

$$A=\left[ \begin{array}{rrr} 2 & 1 & 3 \\ 1 & 5 & -3 \\ 3 & -3 & 7 \end{array} \right] \nonumber$$

Use Definition $\PageIndex{2}$ to show that $A$ is symmetric.

Solution

By Definition $\PageIndex{2}$, we need to show that $A = A^T$. Now, using Definition $\PageIndex{1}$,

$$A^{T} = \left[ \begin{array}{rrr} 2 & 1 & 3 \\ 1 & 5 & -3 \\ 3 & -3 & 7 \end{array} \right]\nonumber$$

Hence, $A = A^{T}$, so $A$ is symmetric.

Example $\PageIndex{3}$: A Skew Symmetric Matrix

Let

$$A=\left[ \begin{array}{rrr} 0 & 1 & 3 \\ -1 & 0 & 2 \\ -3 & -2 & 0 \end{array} \right] \nonumber$$

Show that $A$ is skew symmetric.

Solution

By Definition $\PageIndex{2}$,

$$A^{T} = \left[ \begin{array}{rrr} 0 & -1 & -3\\ 1 & 0 & -2\\ 3 & 2 & 0 \end{array} \right] \nonumber$$

You can see that each entry of $A^T$ is equal to $-1$ times the same entry of $A$. Hence, $A^{T} = - A$ and so by Definition $\PageIndex{2}$, $A$ is skew symmetric.