3 Matrices 3.2 Matrix multiplication 3.4 Multiplication properties

3.3 Transpose

Definition 3.3.1.

Let $A=(a_{ij})$ be a $m\times n$ matrix. The transpose of $A$ , written $A^{T}$ , is the $n\times m$ matrix whose $i, j$ entry is $a_{ji}$ .

You can think of the transpose as being obtained by reflecting $A$ in the south east diagonal starting in the top left hand corner, or as the matrix whose columns are the rows of $A$ , or the matrix whose rows are the columns of $A$ .

Example 3.3.1.

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If $A=\begin{pmatrix}1&2&3\\ 4&5&6\end{pmatrix}$ then $A^{T}=\begin{pmatrix}1&4\\ 2&5\\ 3&6\end{pmatrix}$ .
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If $A=\begin{pmatrix}1&2\\ 3&4\end{pmatrix}$ then $A^{T}=\begin{pmatrix}1&3\\ 2&4\end{pmatrix}$ .
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If $A=\begin{pmatrix}1\\ 2\\ 3\end{pmatrix}$ then $A^{T}=\begin{pmatrix}1&2&3\end{pmatrix}$ .

It’s common to use transposes when we want to think geometrically, because if $\mathbf{x}\in\mathbb{R}^{n}$ then $\mathbf{x}^{T}\mathbf{x}$ is equal to

x_{1}^{2}+x_{2}^{2}+\cdots+x_{n}^{2}

which is the square of the length of $\mathbf{x}$ . (As usual, we have identified the $1\times 1$ matrix $\mathbf{x}^{T}\mathbf{x}$ with a number here).

When $\mathbf{z}$ is a complex column vector, that is, an element of $\mathbb{C}^{n}$ for some $n$ , this doesn’t quite work. If $\mathbf{z}=\begin{pmatrix}1\\ i\end{pmatrix}$ for example, then $\mathbf{z}^{T}\mathbf{z}=0$ , which is not a good measure of the length of $\mathbf{z}$ . For this reason, when people work with complex vectors they often use the conjugate transpose $A^{H}$ defined to be the matrix whose entries are the complex conjugates of the entries of $A^{T}$ . With this definition, for a complex vector $\mathbf{z}=\begin{pmatrix}z_{1}\\ \vdots\\ z_{n}\end{pmatrix}$ we get

\mathbf{z}^{H}\mathbf{z}=|z_{1}|^{2}+\cdots+|z_{n}|^{2}.