m2r (version 1.0.0)

snf: Smith normal form

Description

For an integer matrix M, this computes the matrices D, P, and Q such that D = PMQ, which can be seen as an analogue of the singular value decomposition. All are integer matrices, and P and Q are unimodular (have determinants +- 1).

Usage

snf(mat, code = FALSE)
snf.(mat, code = FALSE)

Arguments

mat

a matrix (integer entries)

code

return only the M2 code? (default: FALSE)

Value

a list of integer matrices D, P, and Q

Examples

 requires Macaulay2

##### basic usage
########################################

M <- matrix(c(
   2,  4,   4,
  -6,  6,  12,
  10, -4, -16
), nrow = 3, byrow = TRUE)

snf(M)

(mats <- snf(M))
P <- mats$P; D <- mats$D; Q <- mats$Q

P %*% M %*% Q                # = D
solve(P) %*% D %*% solve(Q)  # = M

det(P)
det(Q)


M <- matrix(c(
     1,    2,    3,
     1,   34,   45,
  2213, 1123, 6543,
     0,    0,    0
), nrow = 4, byrow = TRUE)
(mats <- snf(M))
P <- mats$P; D <- mats$D; Q <- mats$Q
P %*% M %*% Q                # = D


##### other options
########################################

snf.(M)
snf(M, code = TRUE)