Kronecker(克罗内克)积
如果A是一个 m × n 的矩阵，而B是一个 p × q 的矩阵，克罗内克积A
× B则是一个 mp × nq 的分块矩阵.
在R中使用kronecker来计算两个矩阵的克罗内克积.
例如
[参考]
> x <- matrix(1:10,2,5)
> x
[,1] [,2] [,3] [,4] [,5]
[1,] 1 3 5 7 9
[2,] 2 4 6 8 10
> kronecker(x,x)
[,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] [,11] [,12] [,13] [,14]
[1,] 1 3 5 7 9 3 9 15 21 27 5 15 25 35
[2,] 2 4 6 8 10 6 12 18 24 30 10 20 30 40
[3,] 2 6 10 14 18 4 12 20 28 36 6 18 30 42
[4,] 4 8 12 16 20 8 16 24 32 40 12 24 36 48
[,15] [,16] [,17] [,18] [,19] [,20] [,21] [,22] [,23] [,24] [,25]
[1,] 45 7 21 35 49 63 9 27 45 63 81
[2,] 50 14 28 42 56 70 18 36 54 72 90
[3,] 54 8 24 40 56 72 10 30 50 70 90
[4,] 60 16 32 48 64 80 20 40 60 80 100
x与t(x)的kronecker积.
> kronecker(x,t(x))
[,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10]
[1,] 1 2 3 6 5 10 7 14 9 18
[2,] 3 4 9 12 15 20 21 28 27 36
[3,] 5 6 15 18 25 30 35 42 45 54
[4,] 7 8 21 24 35 40 49 56 63 72
[5,] 9 10 27 30 45 50 63 70 81 90
[6,] 2 4 4 8 6 12 8 16 10 20
[7,] 6 8 12 16 18 24 24 32 30 40
[8,] 10 12 20 24 30 36 40 48 50 60
[9,] 14 16 28 32 42 48 56 64 70 80
[10,] 18 20 36 40 54 60 72 80 90 100[参考]
2. > help(kronecker)
kronecker package:base R Documentation
Kronecker Products on Arrays
Description:
Computes the generalised kronecker product of two arrays, ‘X’ and
‘Y’.
Usage:
kronecker(X, Y, FUN = "*", make.dimnames = FALSE, ...)
X %x% Y
Arguments:
X: A vector or array.
Y: A vector or array.
FUN: a function; it may be a quoted string.
make.dimnames: Provide dimnames that are the product of the dimnames of
‘X’ and ‘Y’.
...: optional arguments to be passed to ‘FUN’.
Details:
If ‘X’ and ‘Y’ do not have the same number of dimensions, the
smaller array is padded with dimensions of size one. The returned
array comprises submatrices constructed by taking ‘X’ one term at
a time and expanding that term as ‘FUN(x, Y, ...)’.
‘%x%’ is an alias for ‘kronecker’ (where ‘FUN’ is hardwired to
‘"*"’).
Value:
An array ‘A’ with dimensions ‘dim(X) * dim(Y)’.
Author(s):
Jonathan Rougier
References:
Shayle R. Searle (1982) _Matrix Algebra Useful for Statistics._
John Wiley and Sons.
See Also:
‘outer’, on which ‘kronecker’ is built and ‘%*%’ for usual matrix
multiplication.
Examples:
# simple scalar multiplication
( M <- matrix(1:6, ncol = 2) )
kronecker(4, M)
# Block diagonal matrix:
kronecker(diag(1, 3), M)
# ask for dimnames
fred <- matrix(1:12, 3, 4, dimnames = list(LETTERS[1:3], LETTERS[4:7]))
bill <- c("happy" = 100, "sad" = 1000)
kronecker(fred, bill, make.dimnames = TRUE)
bill <- outer(bill, c("cat" = 3, "dog" = 4))
kronecker(fred, bill, make.dimnames = TRUE)
