Question

Given a 100×100 matrix, we want to compute a 20×20 matrix, every cell of it represents the mean of a 5×5 square of the original matrix.

How to perform it? (If you have any better name for this operation please comment to rename question).

Answer 1

Here are a couple of options.

The straightforward approach is to use aggregate() from the raster package:

m <- matrix(1:10000, ncol=100)

library(raster)
r <- raster(m)
as.matrix(aggregate(r, 5))

## aggregate() also supports non-square aggregation windows
as.matrix(aggregate(r, c(20, 50)))
#        [,1]   [,2]   [,3]   [,4]   [,5]
# [1,]  975.5 2975.5 4975.5 6975.5 8975.5
# [2,] 1025.5 3025.5 5025.5 7025.5 9025.5

For a more elegant or obfuscated approach (depending on your point of view) use a couple of matrix multiplications:

m <- matrix(1:10000, ncol=100)

mm <- suppressWarnings(matrix(rep(c(1, rep(0, 20)), each=5), ncol=20, nrow=100))
(t(mm) %*% m %*% mm)/25