8.3 Matrix Transpose

The transpose of any (\(r\times c\)) \(\mathbf{A}\) matrix is the (\(c \times r\)) matrix denoted as \(\mathbf{A}^T\) or \(\mathbf{A}'\) such that \(a_{ij}\) is replaced by \(a_{ji}\) everywhere.

A
t(A)

A matrix is square if \(c=r\). The diagonal of a square matrix are the elements of \(a_{ii}\) and the off-diagonal elements of a square matrix are \(a_{ij}\) where \(i\not=j\).

If \(\mathbf{A}\) is symmetric, then \(\mathbf{A} = \mathbf{A}^T\).

For any matrix \(\mathbf{A}\), \(\mathbf{A}^T\mathbf{A}\) will be a square matrix.

8.3.1 Properties

The transpose of a matrix product is \[(\mathbf{AB})^T = \mathbf{B}^T\mathbf{A}^T\] The transpose of a matrix sum is \[(\mathbf{A}+\mathbf{B})^T = \mathbf{A}^T+\mathbf{B}^T\]