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Some notes I'm reading say that:

$$x = \sum_{i=1}^m c_i\langle x,u_i\rangle u_i$$ for all $x\in\mathbb{R}^n$ (where $c_i > 0$ are positive numbers and $u_i$'s are unit vectors) can be equivalently written in matrix (or operator) notation as $$\sum_{i=1}^m c_i u_i\otimes u_i = I_n$$ where $I_n$ is the identity map on $\mathbb{R}^n$, and for any unit vector $u$, $u\otimes u$ is the rank-one orthogonal projection onto the span of $u$, i.e. the map $x\mapsto \langle x,u\rangle u$. The trace of this projection is $1$.

So, I haven't seen the above notation $\otimes$ earlier - and I'm a little confused what it means? Could someone help me understand?

What is the meaning of rank-one orthogonal projection here, and what does the operator really do? How is this equivalent to the first condition?

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It's the Outer/Tensor product and defined by $u \otimes v = uv^\top$ or alternatively, you can think of it as the function

$$ (u \otimes v)(x) = \langle v, x \rangle u, $$

which is of course what you get when you multiply $uv^\top$ and $x$ together.

Note: there is also a Kronecker product which uses the same symbol and the relationship is

$$ u \otimes_{\rm out} v = u \otimes_{\rm kr} v^\top. $$

The Kronecker product of two column vectors $(a_i), (b_j)$ is a vector whose $(in + j)$-th entry is $a_ib_j$. The outer product is a matrix whose $(i,j)$-th entry is $a_ib_j$.

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In particular, if u and v are written as "column" matrices, $u= \begin{pmatrix}u_1 \\ u_2 \\ u_3 \end{pmatrix}$ and $v= \begin{pmatrix}v_1 \\ v_2 \\ v_3 \end{pmatrix}$, then $v^T= \begin{pmatrix}v_1 & v_2 & v_3\end{pmatrix}$, a"row" matrix, and $uv^T= \begin{pmatrix}u_1 \\ u_2 \\ u_3 \end{pmatrix}\begin{pmatrix}v_1 & v_2 & v_3\end{pmatrix}= $$\begin{pmatrix} u_1v_1 & u_1v_2 & u_1v_3 \\ u_2v_1 & u_2v_2 & u_2v_3 \\ u_3v_1 & u_3v_2 & u_3v_3 \end{pmatrix}$, a 3 by 3 matrix.

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