What is the name of the property $a^T(b\odot c)=b^T \text{diag}(a)c$, and what is the definition $b\odot c$?

Question

In one of the research paper I read that $$\mathbb{a}^T(\mathbb{b}\odot \mathbb{c})=\mathbb{b}^T \text{diag}(\mathbb{a})\mathbb{c}$$ where $\mathbb{a}, \mathbb{b},\mathbb{c}$ are vectors of same size. What is the name of above property and what does $\mathbb{b}\odot \mathbb{c}$ means? Thanks in advance. Here is the snap shot of the paper I am reading

Answer 1

The notation ${\rm diag}({\bf a})$ is defined in the paper you linked, page 4027, column 1, last line. I have never used the notation $\odot$ but suspect that it is the elementwise product of two vectors, in which case for ${\bf a}=(a_1,a_2,\ldots,a_n)$ etc we have $$\eqalign{ {\bf a}^T({\bf b}\odot{\bf c}) &=\pmatrix{a_1&\cdots&a_n\cr}\pmatrix{b_1c_1\cr\vdots\cr b_nc_n\cr}\cr &=a_1b_1c_1+\cdots+a_nb_nc_n\cr &=\pmatrix{b_1&\cdots&b_n\cr}\pmatrix{a_1c_1\cr\vdots\cr a_nc_n\cr}\cr &=\pmatrix{b_1&\cdots&b_n\cr}\pmatrix{a_1&\cdots&0\cr\vdots&\ddots&\vdots\cr0&\cdots&a_n\cr}\pmatrix{c_1\cr\vdots\cr c_n\cr}\cr &={\bf b}^T{\rm diag}({\bf a}){\bf c}\ .\cr}$$