pMatrix-class Permutation matrices

Description

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.

Details

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 from the Class

Objects can be created by calls of the form new("pMatrix", ...) or by coercion from an integer permutation vector, see below.

Slots

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.

Extends

Class "indMatrix", directly.

Methods

%*%

signature(x = "matrix", y = "pMatrix") and other signatures (use showMethods("%*%", class="pMatrix")): ...

coerce

signature(from = "integer", to = "pMatrix"): This is enables typical "pMatrix" construction, given a permutation vector of 1:n, see the first example.

coerce

signature(from = "numeric", to = "pMatrix"): a user convenience, to allow as(perm, "pMatrix") for numeric perm with integer values.

coerce

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".)

coerce

signature(from = "pMatrix", to = "ngTMatrix"): coercion to sparse logical matrix of class ngTMatrix.

determinant

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.

solve

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.

t

signature(x = "pMatrix"): return the transpose of the permutation matrix (which is also the inverse of the permutation matrix).

Note

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.

See Also

invPerm(p) computes the inverse permutation of an integer (index) vector p.

Examples

(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, ]