Basic utilities for multi-companion matrices
mc_matrix.RdCompute the dense matrix representation of a multi-companion matrix or convert the argument to an ordinary matrix.
Details
mc_matrix returns an ordinary matrix. It returns x if
x is an ordinary matrix (is.matrix(x) == TRUE), converts
x to a matrix with one row if x is a vector, and returns
as.matrix(x) otherwise. mc_matrix is used by some
functions in package mcompanion that want to allow flexible
format for the top of a multicompanion matrix or even the whole matrix
(e.g. x may be a MultiCompanion object) but are not
really multi-companion aware.
For mc_full, x is normally the top part of a
multi-companion matrix. Rows are appended as necessary to obtain the
dense representation of the matrix and the result is guaranteed to be
a multi-companion matrix. It is an error to have more rows than
columns. If the number of rows is equal to the number of columns,
i.e. x is the whole matrix, the effect is that x is
converted to an ordinary matrix but no check is made to see if the
result is indeed a multi-companion matrix.
x may be a vector if the multi-companion order is 1.
Give the multi-companion order of a square matrix
Determine the multi-companion order of a square matrix or check if a matrix may be the bottom part of a multi-companion matrix.
In mc_order(x) should be a square matrix, while in
is_mc_bottom(x) the matrix is usually rectangular.
The bottom part of a multi-companion matrix is of the form [I 0], where I is an identity matrix and 0 is a matrix of zeroes. The top consists of the rows above the bottom part. The multi-companion order is the number of rows in the top of a multi-companion matrix.
Identity matrices have mc_order zero.
Other general matrices have mc_order equal to the number of
rows. In particular, an \(1\times1\) matrix has mc_order
zero, if its only element is equal to one, and mc_order one
otherwise.
Acordingly, is_mc_bottom(x) returns TRUE if x is the
identity matrix or a matrix with zero rows. This is consistent with
the treatment of the identity matrix as multi-companion of multi order
0 and a general matrix as multi-companion of multi-companion order
equal to the number of its rows.
Value
for mc_full, the multi-companion matrix as an ordinary dense
matrix object.
For mc_matrix, an ordinary matrix.
for mc_order, the multi-companion order of x, a
non-negative integer
for is_mc_bottom, TRUE if x may be the bottom part of a
multi-companion matrix and FALSE otherwise.
References
Boshnakov GN (2002). “Multi-companion matrices.” Linear Algebra Appl., 354, 53--83. ISSN 0024-3795, doi:10.1016/S0024-3795(01)00475-X .
Note
mc_matrix is not multi-companion specific, except that it
converts a vector to a matrix with one row (not column). For square
matrices these functions are not really multi-companion specific.
It may make sense to allow non-square matrices also for mc_order.
Examples
mc <- mCompanion("sim", dim = 4, mo = 2)
mc
#> 4 x 4 Matrix of class "MultiCompanion"
#> [,1] [,2] [,3] [,4]
#> [1,] 0.01366836 -1.144055 0.3276120 -0.7300661
#> [2,] 0.60520825 -1.685660 0.3463819 -0.6725063
#> [3,] 1.00000000 0.000000 0.0000000 0.0000000
#> [4,] 0.00000000 1.000000 0.0000000 0.0000000
mc_order(mc)
#> [1] 2
x <- mc[1:2, ] # the top of mc
x
#> 2 x 4 Matrix of class "dgeMatrix"
#> [,1] [,2] [,3] [,4]
#> [1,] 0.01366836 -1.144055 0.3276120 -0.7300661
#> [2,] 0.60520825 -1.685660 0.3463819 -0.6725063
x2 <- mc[] # whole mc as ordinary matrix
x2
#> 4 x 4 Matrix of class "MultiCompanion"
#> [,1] [,2] [,3] [,4]
#> [1,] 0.01366836 -1.144055 0.3276120 -0.7300661
#> [2,] 0.60520825 -1.685660 0.3463819 -0.6725063
#> [3,] 1.00000000 0.000000 0.0000000 0.0000000
#> [4,] 0.00000000 1.000000 0.0000000 0.0000000
mc_matrix(mc)
#> [,1] [,2] [,3] [,4]
#> [1,] 0.01366836 -1.144055 0.3276120 -0.7300661
#> [2,] 0.60520825 -1.685660 0.3463819 -0.6725063
#> [3,] 1.00000000 0.000000 0.0000000 0.0000000
#> [4,] 0.00000000 1.000000 0.0000000 0.0000000
mc_matrix(x2)
#> [,1] [,2] [,3] [,4]
#> [1,] 0.01366836 -1.144055 0.3276120 -0.7300661
#> [2,] 0.60520825 -1.685660 0.3463819 -0.6725063
#> [3,] 1.00000000 0.000000 0.0000000 0.0000000
#> [4,] 0.00000000 1.000000 0.0000000 0.0000000
## mc_matrix() doesn't append rows to its argument
mc_matrix(x)
#> [,1] [,2] [,3] [,4]
#> [1,] 0.01366836 -1.144055 0.3276120 -0.7300661
#> [2,] 0.60520825 -1.685660 0.3463819 -0.6725063
## mc_full() appends rows, to make the matrix square multicompanion
mc_full(x)
#> [,1] [,2] [,3] [,4]
#> [1,] 0.01366836 -1.144055 0.3276120 -0.7300661
#> [2,] 0.60520825 -1.685660 0.3463819 -0.6725063
#> [3,] 1.00000000 0.000000 0.0000000 0.0000000
#> [4,] 0.00000000 1.000000 0.0000000 0.0000000
## mc and x2 are square, so not amended:
mc_full(mc)
#> [,1] [,2] [,3] [,4]
#> [1,] 0.01366836 -1.144055 0.3276120 -0.7300661
#> [2,] 0.60520825 -1.685660 0.3463819 -0.6725063
#> [3,] 1.00000000 0.000000 0.0000000 0.0000000
#> [4,] 0.00000000 1.000000 0.0000000 0.0000000
mc_full(x2)
#> [,1] [,2] [,3] [,4]
#> [1,] 0.01366836 -1.144055 0.3276120 -0.7300661
#> [2,] 0.60520825 -1.685660 0.3463819 -0.6725063
#> [3,] 1.00000000 0.000000 0.0000000 0.0000000
#> [4,] 0.00000000 1.000000 0.0000000 0.0000000
## a vector argument is treated as a matrix with 1 row:
mc_matrix(1:4)
#> [,1] [,2] [,3] [,4]
#> [1,] 1 2 3 4
mc_full(1:4)
#> [,1] [,2] [,3] [,4]
#> [1,] 1 2 3 4
#> [2,] 1 0 0 0
#> [3,] 0 1 0 0
#> [4,] 0 0 1 0
## mc_order(1:4) # not by mc_order
m <- mCompanion(matrix(1:8, nrow = 2))
mc_matrix(m)
#> [,1] [,2] [,3] [,4]
#> [1,] 1 3 5 7
#> [2,] 2 4 6 8
#> [3,] 1 0 0 0
#> [4,] 0 1 0 0
mc_order(m)
#> [1] 2
m[-c(1,2), ]
#> 2 x 4 Matrix of class "dgeMatrix"
#> [,1] [,2] [,3] [,4]
#> [1,] 1 0 0 0
#> [2,] 0 1 0 0
is_mc_bottom(m[-c(1,2), ]) # TRUE
#> [1] TRUE
## TRUE for reactangular diagonal matrix with nrow < ncol
is_mc_bottom(diag(1, nrow = 3, ncol = 5))
#> [1] TRUE
## border cases
is_mc_bottom(matrix(0, nrow = 0, ncol = 4)) # TRUE, 0 rows
#> [1] TRUE
is_mc_bottom(diag(4)) # TRUE, square diagonal matrix
#> [1] TRUE