I'm working through a paper on Alternating Least Squares and I have come across a notation that I do not understand:

$$r_{ij} = < u_i , m_j >, ∀ i, j$$

Can anyone help by breaking this down step by step for me?

I have searched online for a description of this notation but I can't find an explanation.

As a first step to answering the question, it looks like the meaning of the angle brackets are as follows:

$${\mathcal {E}}_{n}=\langle {\vec {e}}_{1},\,\ldots ,\,{\vec {e}}_{n}\rangle \qquad standard \; basis \; for \; ${\displaystyle \mathbb {R} ^{n}} \mathbb {R} ^{n}$$

Source: https://en.wikibooks.org/wiki/Linear_Algebra/Notation


$\langle{u_i,m_j}\rangle$ is just an inner product on a vector space $\mathcal{V}$, i.e. a function $\langle{u_i,m_j}\rangle: \mathcal{V} \times \mathcal{V} \to \mathbb{R}, $ which follows 3 axioms (linearity in $u_i$,symmetric,positive definite). Therefore $r_{i,j}$ is just a scalar value. This is essentially the dot product as @Kristina says but in a more generalised form for vector spaces.


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