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)` (`\approx` `t(P) %*% M` ) | = | `M[i(p), ]` |

`MP'` | = | `tcrossprod(M,P)` (`\approx` `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")`

): ...- 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).

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, "TsparseMatrix")
p10[1:7, 1:4] # gives an "ngTMatrix" (most economic!)
## row-indexing of a <pMatrix> keeps it as an <indMatrix>:
p10[1:3, ]
```

[Package *Matrix* version 1.5-1 Index]