| pMatrix-class {Matrix} | R Documentation |
The "pMatrix" class is the class of permutation
matrices, stored as 1-based integer permutation vectors.
Matrix (vector) multiplication with permutation matrices is equivalent to row or column permutation, and is implemented that way in the Matrix package, see the ‘Details’ below.
Matrix multiplication with permutation matrices is equivalent to row or column permutation. Here are the four different cases for an arbitrary matrix M and a permutation matrix P (where we assume matching dimensions):
| MP | = | M %*% P | = | M[, i(p)] |
| PM | = | P %*% M | = | M[ p , ] |
| P'M | = | crossprod(P,M) (~=t(P) %*% M) | = | M[i(p), ] |
| MP' | = | tcrossprod(M,P) (~=M %*% t(P)) | = | M[ , p ] |
where p is the “permutation vector” corresponding to the
permutation matrix P (see first note), and i(p) is short
for invPerm(p).
Also one could argue that these are really only two cases if you take
into account that inversion (solve) and transposition
(t) are the same for permutation matrices P.
Objects can be created by calls of the form new("pMatrix", ...)
or by coercion from an integer permutation vector, see below.
perm:An integer, 1-based permutation vector, i.e.
an integer vector of length Dim[1] whose elements form a
permutation of 1:Dim[1].
Dim:Object of class "integer". The dimensions
of the matrix which must be a two-element vector of equal,
non-negative integers.
Dimnames:list of length two; each component
containing NULL or a character vector length
equal the corresponding Dim element.
Class "indMatrix", directly.
signature(x = "matrix", y = "pMatrix") and other
signatures (use showMethods("%*%", class="pMatrix")): ...
signature(from = "integer", to = "pMatrix"):
This is enables typical "pMatrix" construction, given
a permutation vector of 1:n, see the first example.
signature(from = "numeric", to = "pMatrix"):
a user convenience, to allow as(perm, "pMatrix") for
numeric perm with integer values.
signature(from = "pMatrix", to = "matrix"):
coercion to a traditional FALSE/TRUE matrix of
mode logical.
(in earlier version of Matrix, it resulted in a 0/1-integer
matrix; logical makes slightly more sense, corresponding
better to the “natural” sparseMatrix counterpart,
"ngTMatrix".)
signature(from = "pMatrix", to = "ngTMatrix"):
coercion to sparse logical matrix of class ngTMatrix.
signature(x = "pMatrix", logarithm="logical"):
Since permutation matrices are orthogonal, the determinant must be
+1 or -1. In fact, it is exactly the sign of the
permutation.
signature(a = "pMatrix", b = "missing"): return
the inverse permutation matrix; note that solve(P) is
identical to t(P) for permutation matrices. See
solve-methods for other methods.
signature(x = "pMatrix"): return the transpose of
the permutation matrix (which is also the inverse of the
permutation matrix).
For every permutation matrix P, there is a corresponding
permutation vector p (of indices, 1:n), and these are related by
P <- as(p, "pMatrix") p <- P@perm
see also the ‘Examples’.
“Row-indexing” a permutation matrix typically returns
an "indMatrix". See "indMatrix" for all other
subsetting/indexing and subassignment (A[..] <- v) operations.
invPerm(p) computes the inverse permutation of an
integer (index) vector p.
(pm1 <- as(as.integer(c(2,3,1)), "pMatrix"))
t(pm1) # is the same as
solve(pm1)
pm1 %*% t(pm1) # check that the transpose is the inverse
stopifnot(all(diag(3) == as(pm1 %*% t(pm1), "matrix")),
is.logical(as(pm1, "matrix")))
set.seed(11)
## random permutation matrix :
(p10 <- as(sample(10),"pMatrix"))
## Permute rows / columns of a numeric matrix :
(mm <- round(array(rnorm(3 * 3), c(3, 3)), 2))
mm %*% pm1
pm1 %*% mm
try(as(as.integer(c(3,3,1)), "pMatrix"))# Error: not a permutation
as(pm1, "ngTMatrix")
p10[1:7, 1:4] # gives an "ngTMatrix" (most economic!)
## row-indexing of a <pMatrix> keeps it as an <indMatrix>:
p10[1:3, ]