I have a formula for which I'm not 100% sure whether I interpret it correctly. Apologies, if the question is trivial. The formula is as follows:
$$ f(w, R) = \max\bigg\{{\sum_{s = 1}^k w_s \cdot r_{(i,j),s}: i,j \in \{1, \dots, d\}\bigg\}} $$
$R$ is a set $s$-many of matrices of dimension $d \times d$, each matrix $r_{(i,j),s}$ in the set is identified by the index $s$, the elements of the matrix are indexed by $i$ and $j$ (row and column). $w$ is a vector of weights. ($max$ is the maximum function)
My interpretation of the formula is as follows:
- Each matrix $r$ ($s = 1$ to $k$) is multiplied with the corresponding element of the vector $w$, i.e., each element of matrix $r_s$ in multiplied by $w_s$
- all matrices $r$ ($s = 1$ to $k$) are added elementwise, resulting in a single matrix $p$ of size $i \times j$ ($p_{(i,j)} = r_{(i,j),1} + r_{(i,j),2} + \dots r_{(i,j),k} \;for \; i = 1 \dots d \; and \; j = 1 \dots d$
- From the resulting matrix the larges element is selected by the max-function, i.e., the result of this formula is a single number
My question is: Is this interpretation correct or is there another way to interpret the formula?
What happens if the set of matrices in the input $R$ is larger than $d \times d\,$? Is the result still a single value?
Many thanks in advanced!