Matrix Transpose

The transpose of a matrix is an operation that flips the matrix over its main diagonal. This switches the row and column indices of the matrix.

If we have a matrix \(A\), its transpose is denoted as \(A^T\).

Example

\[A = \begin{bmatrix} a & b \\ c & d \\ e & f \end{bmatrix} \quad \implies \quad A^T = \begin{bmatrix} a & c & e \\ b & d & f \end{bmatrix}\]

The element at row i, column j of the transpose is the element from row j, column i of the original matrix: \((A^T)_{ij} = A_{ji}\).

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Transpose of a Matrix Product

A key property of transposes involves the product of two matrices. The transpose of a matrix product is the product of their individual transposes in reverse order.

\[(A B)^T = B^T A^T\]

Proof by Example

Let’s verify this. First, we calculate \(AB\) and then find its transpose \((AB)^T\).

\[AB = \begin{bmatrix} a & b \\ c & d \end{bmatrix} \begin{bmatrix} e & f \\ g & h \end{bmatrix} = \begin{bmatrix} ae+bg & af+bh \\ ce+dg & cf+dh \end{bmatrix} \quad \implies \quad (AB)^T = \begin{bmatrix} ae+bg & ce+dg \\ af+bh & cf+dh \end{bmatrix}\]

Next, we find the transposes of \(A\) and \(B\) individually and multiply them in reverse order, \(B^T A^T\).

\[B^T A^T = \begin{bmatrix} e & g \\ f & h \end{bmatrix} \begin{bmatrix} a & c \\ b & d \end{bmatrix} = \begin{bmatrix} ea+gb & ec+gd \\ fa+hb & fc+hd \end{bmatrix} = \begin{bmatrix} ae+bg & ce+dg \\ af+bh & cf+dh \end{bmatrix}\]

Cool, The results are identical, confirming that \((AB)^T = B^T A^T\).