| Type: | Package |
| Title: | Exact Matrix Algebra for Rational Matrices |
| Version: | 1.0.0 |
| Maintainer: | Stéphane Laurent <laurent_step@outlook.fr> |
| Description: | Provides functions to deal with matrix algebra for matrices with rational entries: determinant, rank, image and kernel, inverse, Cholesky decomposition. All computations are exact. |
| License: | GPL-3 |
| URL: | https://github.com/stla/RationalMatrix |
| BugReports: | https://github.com/stla/RationalMatrix/issues |
| Imports: | gmp, Rcpp (≥ 1.0.9) |
| LinkingTo: | BH, Rcpp, RcppEigen |
| Encoding: | UTF-8 |
| RoxygenNote: | 7.2.3 |
| SystemRequirements: | C++ 17, gmp |
| NeedsCompilation: | yes |
| Packaged: | 2023-01-27 13:10:50 UTC; stla |
| Author: | Stéphane Laurent [aut, cre] |
| Repository: | CRAN |
| Date/Publication: | 2023-01-27 16:50:09 UTC |
'UtDU' decomposition of a rational matrix
Description
Cholesky-'UtDU' decomposition of a symmetric rational matrix.
Usage
QcholUtDU(M)
Arguments
M |
a square matrix such that |
Value
The Cholesky-'UtDU' decomposition of M in a list
(see example).
Note
Symmetry is not checked! Only the lower triangular part of
M is used.
Examples
library(RationalMatrix)
x <- matrix(c(1:5, (1:5)^2), 5, 2)
x <- cbind(x, x[, 1L] + 3L*x[, 2L])
M <- crossprod(x)
UtDU <- QcholUtDU(M)
library(gmp)
U <- as.bigq(UtDU$U)
D <- matrix("0", 3L, 3L)
diag(D) <- UtDU$D
D <- as.bigq(D)
perm <- UtDU$perm
UP <- U[, perm]
t(UP) %*% D %*% UP # this is `M`
Determinant of a rational matrix
Description
Determinant of a square matrix with rational entries.
Usage
Qdet(M)
Arguments
M |
a square matrix such that |
Value
A string: quoted rational number representing the determinant.
Examples
library(RationalMatrix)
M <- cbind(c("1/2", "3"), c("5/3", "-2/7"))
Qdet(M)
Inverse of a rational matrix
Description
Inverse matrix of a square rational matrix.
Usage
Qinverse(M)
Arguments
M |
a square matrix such that |
Value
A character matrix representing the inverse of M.
Examples
library(RationalMatrix)
M <- cbind(c("1/2", "3", "1"), c("5/3", "-2/7", "10/3"), c("0", "1", "2"))
Qinverse(M)
Check injectivity
Description
Checks whether a rational matrix represents an injective linear map (i.e. has trivial kernel).
Usage
QisInjective(M)
Arguments
M |
a matrix such that |
Value
A Boolean value indicating whether the linear map corresponding to
M is injective.
Examples
library(RationalMatrix)
set.seed(666L)
M <- matrix(rpois(35L, 1), 5L, 7L)
QisInjective(M)
Check invertibility
Description
Checks whether a square rational matrix is invertible.
Usage
QisInvertible(M)
Arguments
M |
a square matrix such that |
Value
A Boolean value indicating whether M is invertible.
Examples
library(RationalMatrix)
set.seed(666L)
M <- matrix(rpois(25L, 1), 5L, 5L)
QisInvertible(M)
Check surjectivity
Description
Checks whether a rational matrix represents a surjective linear map.
Usage
QisSurjective(M)
Arguments
M |
a matrix such that |
Value
A Boolean value indicating whether the linear map corresponding to
M is surjective.
Examples
library(RationalMatrix)
set.seed(666L)
M <- matrix(rpois(35L, 1), 7L, 5L)
QisSurjective(M)
Kernel of a rational matrix
Description
Kernel (null-space) of a rational matrix.
Usage
Qkernel(M)
Arguments
M |
a matrix such that |
Value
A character matrix representing a basis of the kernel of M.
Note that this basis is not orthogonal.
Examples
library(RationalMatrix)
set.seed(666L)
M <- matrix(rpois(30L, 6), 10L, 3L)
M <- cbind(M, M[,1] + M[,2], M[,2] + 2L*M[,3])
Qkernel(M)
Range of a rational matrix
Description
Range (column-space, image, span) of a rational matrix.
Usage
Qrange(M)
Arguments
M |
a matrix such that |
Value
A character matrix representing a basis of the range of M.
Note that this basis is not orthogonal.
Examples
library(RationalMatrix)
set.seed(666L)
M <- matrix(rpois(15L, 6), 3L, 5L)
Qrange(M)
Rank of a rational matrix
Description
Returns the rank of a rational matrix.
Usage
Qrank(M)
Arguments
M |
a matrix such that |
Value
An integer, the rank of M.
Examples
library(RationalMatrix)
M <- cbind(c("1/2", "3", "1"), c("5/3", "-2/7", "10/3"), c("1", "1", "2"))
Qrank(M)