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.