# Aggregation Matrix

## Definition

Aggregation Matrix (also Summation Matrix) in the context of Input-Output Analysis is a Boolean Matrix that aims to produce a coarse-grained version of a more granular Input-Output Model.

Aggregation can be for example along sectoral or regional dimensions.

Vectors and Matrices can be aggregated by multiplying with the aggregation matrix. Mathematically a aggregation matrix S is a ${\displaystyle K\times N}$ matrix, where each value ${\displaystyle s_{mn}}$ is either zero or one. The aggregation matrix has in total N non-zero values.

${\displaystyle S=\left({\begin{matrix}s_{00}&s_{01}&\dots &s_{0n}&\dots &s_{0N}\\s_{10}&s_{11}&\dots &s_{1n}&\dots &s_{1N}\\\vdots &\vdots &\ddots &\vdots &\ddots &\vdots \\s_{m0}&s_{m1}&\dots &s_{mn}&\dots &s_{mN}\\\vdots &\vdots &\ddots &\vdots &\ddots &\vdots \\s_{K0}&s_{K1}&\dots &s_{Kn}&\dots &s_{KN}\\\end{matrix}}\right).}$

An aggregation matrix with K rows and N columns is used to aggregate a dimension of size N into a smaller dimension of size K. The N non-zero (unit) values select and group the elements of the vector or matrix that is to be aggregated.

### Vector Quantity Aggregation

A vector Y of dimension N is aggregated into a vector K through its pre-multiplication with the aggregation matrix.

${\displaystyle \mathbf {Y} ^{s}=\mathbf {S} \mathbf {Y} }$

or more explicitly:

${\displaystyle Y_{i}^{s}=\sum _{j=1}^{N}s_{ij}Y_{j}}$

### Matrix Quantity Aggregation

A matrix of dimension N x N is aggregated into a K x K matrix through its pre-multiplication with the aggregation K x N matrix and the post-multiplication with the (N x K) aggregation matrix transpose.

${\displaystyle \mathbf {A} _{s}=\mathbf {S} \mathbf {A} \mathbf {S} ^{T}}$

or more explicitly:

${\displaystyle a_{ij}^{s}=\sum _{k=1}^{N}\sum _{l=1}^{N}s_{ik}a_{lk}s_{il}}$