# What is this matrix/operator notation $\otimes$?

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?

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 $$a_ib_j$$.

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.