9 Matrix Transposition

Transposition is another operation that can also be carried out on matrices. Just as when we transpose a vector, we transpose a matrix by taking each column in turn and making it a row. In other words, we interchange each column and row, so the first column becomes the first row, the second column becomes the second row, etc. For example,

\[ \mathbf{A} = \begin{bmatrix} 112 & 86 & 0 \\ 134 & 94 & 0 \end{bmatrix} \qquad \mathbf{A}^\intercal = \begin{bmatrix} 112 & 134 \\86 & 94 \\ 0 & 0 \end{bmatrix} \]

Formally, if A is an \(n \times k\) matrix with elements \(a_{ij}\), then the transpose of A, denoted \(\mathbf{A}^\intercal\) is a \(k \times n\) matrix where element \(a^\intercal_{ij}=a_{ji}\). Some properties of the matrix transpose are:

\((\mathbf{A}^{\intercal}) ^{^{\intercal}} = \mathbf{A}\)
\((\lambda\mathbf{A})^{^{\intercal}} = \lambda \mathbf{A}^{\intercal}\) where \(\lambda\) is a scalar
\((\mathbf{A} + \mathbf{B})^{^{\intercal}} = \mathbf{A}^{\intercal} + \mathbf{B}^{\intercal}\) (This property extends to more than two matrices; \((\mathbf{A} + \mathbf{B} + \mathbf{C})^{^{\intercal}} = \mathbf{A}^{\intercal} + \mathbf{B}^{\intercal} + \mathbf{C}^{\intercal}\).)
\((\mathbf{A}\mathbf{B})^{^{\intercal}} = \mathbf{B}^{\intercal} \mathbf{A}^{\intercal}\) (This property also extends to more than two matrices; \((\mathbf{A}\mathbf{B}\mathbf{C})^{^{\intercal}} = \mathbf{C}^{\intercal}\mathbf{B}^{\intercal} \mathbf{A}^{\intercal}\).)

Computationally, the t() function will produce the transpose of a matrix in R.

# Create A
A = matrix(
  data = c(112, 134, 86, 94, 0, 0),
  nrow = 2
)

# Display A
A

     [,1] [,2] [,3]
[1,]  112   86    0
[2,]  134   94    0

# Compute transpose
t(A)

     [,1] [,2]
[1,]  112  134
[2,]   86   94
[3,]    0    0

Exercises

Consider the following matrices:

\[ \mathbf{A} = \begin{bmatrix}1 & 5 & -1 \\ 3 & 2 & -1 \\ 0 & 1 & 5 \end{bmatrix} \qquad \mathbf{B} = \begin{bmatrix}3 & 1 & 6 \\ 2 & 0 & 1 \\ -7 & -1 & 2 \end{bmatrix} \]

Verify that \((\mathbf{A}+\mathbf{B})^\intercal = \mathbf{A}^\intercal + \mathbf{B}^\intercal\)?

Verify that \((\mathbf{A}\mathbf{B})^\intercal = \mathbf{B}^\intercal \mathbf{A}^\intercal\)?.