# [R] how to calculate left kronecker product?

Michael Friendly friendly at yorku.ca
Sat Feb 23 20:46:53 CET 2013

For an application, I have formulas defined in terms of a left Kronecker
product of matrices,
A,B, meaning
A \otimes_L B = {A * B[i,j]}  -- matrix on the left multiplies each
element on the right.

The standard kronecker() function is the right Kronecker product,
A \otimes_R B = {A[i,j] * B}  -- matrix on the right multiplies each
element on the left.

The example below shows the result of kronecker() and what I want, but
kronecker() is now defined in generic S4 methods, and  I can't see
how to use more basic functions to get the result I want.  Or, alternatively
how to transform the result of kronecker() to give my wanted.

---- test code ---
A <- matrix(1:4, 2,2, dimnames=list(c("a1", "a2"), c("a1","a2")))
B <- diag(2)
dimnames(B) <- list(c("b1", "b2"), c("b1","b2"))
# standard, right kronecker product: each A[i,j] * B
kronecker(A, B, make.dimnames=TRUE)

# left kronecker product: A * each B[i,j]
wanted <- rbind(
cbind(A * B[1,1], A*B[1,2]),
cbind(A * B[2,1], A*B[2,2]))
rownames(wanted) <- colnames(wanted) <- paste(rep(c("b1", "b2"),
each=2), rownames(wanted), sep=":")
wanted

---- R output ----
> A <- matrix(1:4, 2,2, dimnames=list(c("a1", "a2"), c("a1","a2")))
> B <- diag(2)
> dimnames(B) <- list(c("b1", "b2"), c("b1","b2"))
> # standard, right kronecker product: each A[i,j] * B
> kronecker(A, B, make.dimnames=TRUE)
a1:b1 a1:b2 a2:b1 a2:b2
a1:b1     1     0     3     0
a1:b2     0     1     0     3
a2:b1     2     0     4     0
a2:b2     0     2     0     4
>
> # left kronecker product: A * each B[i,j]
> wanted <- rbind(
+ cbind(A * B[1,1], A*B[1,2]),
+ cbind(A * B[2,1], A*B[2,2]))
>
> rownames(wanted) <- colnames(wanted) <- paste(rep(c("b1", "b2"),
each=2), rownames(wanted), sep=":")
> wanted
b1:a1 b1:a2 b2:a1 b2:a2
b1:a1     1     3     0     0
b1:a2     2     4     0     0
b2:a1     0     0     1     3
b2:a2     0     0     2     4
>

--
Michael Friendly     Email: friendly AT yorku DOT ca
Professor, Psychology Dept. & Chair, Quantitative Methods
York University      Voice: 416 736-2100 x66249 Fax: 416 736-5814
4700 Keele Street    Web:   http://www.datavis.ca