Fat Diagonals

Bastiaan Quast

2022-06-19

Fat diagonal matrices occur when we combine two dimensions of a data set along one edge of a matrix. For example, trade-flow data in the decompr and gvc package have each country-industry combination occur on each edge of the matrix.

A fat diagonal matrix looks like this.

library(diagonals)

## 
## D I
## A G
##     O N
##     A L

fatdiag(12, steps=3)

##       [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] [,11] [,12]
##  [1,]    1    1    1    1    0    0    0    0    0     0     0     0
##  [2,]    1    1    1    1    0    0    0    0    0     0     0     0
##  [3,]    1    1    1    1    0    0    0    0    0     0     0     0
##  [4,]    1    1    1    1    0    0    0    0    0     0     0     0
##  [5,]    0    0    0    0    1    1    1    1    0     0     0     0
##  [6,]    0    0    0    0    1    1    1    1    0     0     0     0
##  [7,]    0    0    0    0    1    1    1    1    0     0     0     0
##  [8,]    0    0    0    0    1    1    1    1    0     0     0     0
##  [9,]    0    0    0    0    0    0    0    0    1     1     1     1
## [10,]    0    0    0    0    0    0    0    0    1     1     1     1
## [11,]    0    0    0    0    0    0    0    0    1     1     1     1
## [12,]    0    0    0    0    0    0    0    0    1     1     1     1

The workhorse function of this package is the fatdiag function, which tries behave similarly to the diag function from the base package, but then with diagonals being fat.

We can also use the function to assign values to the diagonal.

( m <- matrix(111, nrow=6, ncol=9) )

##      [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9]
## [1,]  111  111  111  111  111  111  111  111  111
## [2,]  111  111  111  111  111  111  111  111  111
## [3,]  111  111  111  111  111  111  111  111  111
## [4,]  111  111  111  111  111  111  111  111  111
## [5,]  111  111  111  111  111  111  111  111  111
## [6,]  111  111  111  111  111  111  111  111  111

fatdiag(m, steps=3) <- 5
m

##      [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9]
## [1,]    5    5    5  111  111  111  111  111  111
## [2,]    5    5    5  111  111  111  111  111  111
## [3,]  111  111  111    5    5    5  111  111  111
## [4,]  111  111  111    5    5    5  111  111  111
## [5,]  111  111  111  111  111  111    5    5    5
## [6,]  111  111  111  111  111  111    5    5    5

As can be seen from the above example, the blocks and matrices do not have to be square.

The diagonal of a matrix can also be extracted.

fatdiag(m, steps=3)

##  [1] 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5

We can also specify the size of the block in stead of the number of steps.

fatdiag(12, size=4)

##       [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] [,11] [,12]
##  [1,]    1    1    1    1    0    0    0    0    0     0     0     0
##  [2,]    1    1    1    1    0    0    0    0    0     0     0     0
##  [3,]    1    1    1    1    0    0    0    0    0     0     0     0
##  [4,]    1    1    1    1    0    0    0    0    0     0     0     0
##  [5,]    0    0    0    0    1    1    1    1    0     0     0     0
##  [6,]    0    0    0    0    1    1    1    1    0     0     0     0
##  [7,]    0    0    0    0    1    1    1    1    0     0     0     0
##  [8,]    0    0    0    0    1    1    1    1    0     0     0     0
##  [9,]    0    0    0    0    0    0    0    0    1     1     1     1
## [10,]    0    0    0    0    0    0    0    0    1     1     1     1
## [11,]    0    0    0    0    0    0    0    0    1     1     1     1
## [12,]    0    0    0    0    0    0    0    0    1     1     1     1

This also gives us flexibility in terms of having non-square blocks (and consequently matrices).

fatdiag(12, size=c(3,4) )

##       [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] [,11] [,12]
##  [1,]    1    1    1    1    0    0    0    0    0     0     0     0
##  [2,]    1    1    1    1    0    0    0    0    0     0     0     0
##  [3,]    1    1    1    1    0    0    0    0    0     0     0     0
##  [4,]    0    0    0    0    1    1    1    1    0     0     0     0
##  [5,]    0    0    0    0    1    1    1    1    0     0     0     0
##  [6,]    0    0    0    0    1    1    1    1    0     0     0     0
##  [7,]    0    0    0    0    0    0    0    0    1     1     1     1
##  [8,]    0    0    0    0    0    0    0    0    1     1     1     1
##  [9,]    0    0    0    0    0    0    0    0    1     1     1     1