Extracting all Square Matrices of a Matrix - r

Question

I am trying to extract all possible square matrices of a matrix,
for example I have this matrix:
S = matrix(1:12, nrow=3)
and I want to extract all possible square matrices from S like the following two (3*3) matrices without modifying the structure of the matrix (keeping the order of rows and columns intact):
I1 = matrix(1:9, nrow=3)
I2 = matrix(4:12, nrow=3)
Thanks

Answer 1

The following should do what you want. First some setup.
# Your data
S <- matrix(1:12, nrow=3)
# Set some helpful variables
n <- nrow(S)
m <- ncol(S)
r <- seq_len(min(n, m)) # Sizes of square submatrices to extract
# Number of sq. submatrices for each r element
r.combs <- structure(choose(n, r)*choose(m, r), names = r)
print(r.combs)
# 1 2 3
#12 18 4
# Total number of square submatrices
sum(r.combs)
#[1] 34
So we expect 34 square submatrices of which 12 are 1x1, 18 are 2x2, and 4 are 3x3.
Next, we loop over all square matrices possible r and all combinations
# Initialize list to hold lists of matrices for each R
res <- structure(vector("list", length(r)), names = paste0("r", r))
for (R in r) {
tmp <- list()
R_n <- combn(n, R, simplify = FALSE) # List all combinations in (n choose R)
R_m <- combn(m, R, simplify = FALSE) # List all combinations in (m choose R)
for(i in seq_along(R_n)) {
for (j in seq_along(R_m)){
tmp <- c(tmp, list(S[R_n[[i]], R_m[[j]], drop = FALSE]))
}
}
res[[R]] <- tmp
}
# See structure
str(res, max.level = 1) # See also str(res)
#List of 3
# $ r1:List of 12
# $ r2:List of 18
# $ r3:List of 4
As seen we have the correct number of submatrices for each size.
Edit:
If you want only submatrices that are "directly" present (rows and columns should be adjacent):
res2 <- structure(vector("list", length(r)), names = paste0("r", r))
for (R in r) {
tmp <- list()
for (i in R:n - R) {
for (j in R:m - R) {
tmp <- c(tmp, list(S[i + 1:R, j + 1:R, drop = FALSE]))
}
}
res2[[R]] <- tmp
}
str(res2, max.level = 1)
#List of 3
# $ r1:List of 12
# $ r2:List of 6
# $ r3:List of 2
With strong inspiration form here.