If $u$ and $v$ are in $\Bbb R^n$, how are $u^Tv$ and $v^Tu$ related? How are $uv^T$ and $vu^T$ related?

If $u

jeffp

Answered by Aaron 2 years ago

we have $u,v \in \Bbb R^n$ with

$v=\begin{bmatrix}v_1 \\ v_2 \\ \vdots \\ v_n\end{bmatrix}$ and $v^T=\begin{bmatrix}v_1 & v_2 & \ldots & v_n\end{bmatrix}$

$u=\begin{bmatrix}u_1 \\ u_2 \\ \vdots \\ u_n\end{bmatrix}$ and $u^T=\begin{bmatrix}u_1 & u_2 & \ldots & u_n\end{bmatrix}$

**how are $u^tv$ and $v^tu$ related**

$$u^Tv=\begin{bmatrix}u_1 & u_2 & \ldots & u_n\end{bmatrix} \begin{bmatrix}v_1 \\ v_2 \\ \vdots \\ v_n\end{bmatrix}=u_1v_1+u_2v_2+\ldots+u_nv_n$$

$$v^Tu=\begin{bmatrix}v_1 & v_2 & \ldots & v_n\end{bmatrix} \begin{bmatrix}u_1 \\ u_2 \\ \vdots \\ u_n\end{bmatrix}=u_1v_1+u_2v_2+\ldots+u_nv_n$$

so $u^Tv=v^Tu$

**How are $uv^T$ and $vu^T$ related**

$$uv^T=\begin{bmatrix}u_1 \\ u_2 \\ \vdots \\ u_n\end{bmatrix} \begin{bmatrix}v_1 & v_2 & \ldots & v_n\end{bmatrix} = \begin{bmatrix}u_1v_1 & u_1v_2 & \ldots & u_1v_n \\ u_2v_1 & u_2v_2 & \ldots & u_2v_n \\ \vdots & & \ddots & \\ u_nv_1 & u_nv_2 & & u_nv_n \end{bmatrix}$$

$$vu^T=\begin{bmatrix}v_1 \\ v_2 \\ \vdots \\ v_n\end{bmatrix} \begin{bmatrix}u_1 & u_2 & \ldots & u_n\end{bmatrix} = \begin{bmatrix}v_1u_1 & v_1u_2 & \ldots & v_1u_n \\ v_2u_1 & v_2u_2 & \ldots & v_2u_n \\ \vdots & & \ddots & \\ v_nu_1 & v_nu_2 & & v_nu_n \end{bmatrix}$$

so $uv^T=(vu^T)^T$

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Asked: 2 years ago