decorrelating detector
It all works out because $[\mathbf{R}^{-1}]_{1, i}$ is just a number. It is the number in row $1$ column $i$ of the matrix $\mathbf{R}^{-1}$. So the product $[\mathbf{R}^{-1}]_{1, i}\mathbf{s}_i$ returns a vector of length $N \times 1$. See the line after Equation 10.