Transpose of a Matrix

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To transpose a matrix, means to make rows of a matrix to columns of the transposed matrix (or similarly columns of a matrix to the rows of the transposed matrix). For example \begin{align} \mathbf{A}^T=\begin{bmatrix} 1 & 3\\ 2 & 3 \\ 4 & 1 \end{bmatrix}^T=\begin{bmatrix} 1 & 2 & 4\\ 3& 3& 1 \end{bmatrix} \end{align} In a neat compact notation this means that the $ij$-entry (row $i$, column $j$) of $\mathbf{A}^T$ is the same as the entry $\mathbf{A}_{ji}$ (row $j$, column $i$) \begin{align} (\mathbf{A}^T)_{ij}=\mathbf{A}_{ji} \end{align}

Matrices called symmetric matrices have the property that they equal their transposes \begin{align} \mathbf{A}_{ij}=\mathbf{A}_{ji} \end{align} For example the matrix \begin{align} \begin{bmatrix} 3 & 1 & 7\\ 1 & 2 & 9 \\ 7 & 9 & 4 \end{bmatrix}^T \end{align} is symmetric. If we change rows to columns we get the same matrix back again.


Transpose of Products

If we transpose a product of matrices we have to change the order of multiplication \begin{align} (\mathbf{A}\mathbf{B})^T=\mathbf{B}^T\mathbf{A}^T \end{align}

When we multiply $\mathbf{A}^T\mathbf{A}$ we always get a symmetric matrix, even when $\mathbf{A}$ is no square matrix. For the above matrix $\mathbf{A}$ we have

\begin{align} \mathbf{A}\mathbf{A}^T=\begin{bmatrix} 1 & 3\\ 2 & 3 \\ 4 & 1 \end{bmatrix}\begin{bmatrix} 1 & 2 & 4\\ 3& 3& 1 \end{bmatrix}=\begin{bmatrix} 10 & 11 & 7\\ 11& 13& 11 \\ 7 & 11& 17\end{bmatrix} \end{align} which is symmetric. It's easy to proof that this is in general true \begin{align} (\mathbf{A}\mathbf{A}^T)^T=\mathbf{A}^{TT}\mathbf{A}^T=\mathbf{A}\mathbf{A}^T \end{align} So $\mathbf{A}\mathbf{A}^T$ is symmetric.




Video Lectures:

  • Gilbert Strang - Introduction to Linear Algebra Lec. 5