Companion Matrix of a Polynomial Matrix

Computes a companion matrix for a square (\(m,m)\)-dimensional), matrix polynomial $$a(z) = a_0 + a_1 z + \cdots + a_p z^p$$ The companion matrix is e.g. used to determine the zeroes of a polynomial matrix, see zeroes.
Note that the function throws an error, if the constant term \(a_0\) is singular. There is no check whether some of the leading coefficients are zero. So the results is an \((mp,mp)\)-dimensional matrix, even if \(a_p\) is zero.

Usage

companion_matrix(a)

Arguments

a

A square polynomial matrix, i.e. an object of class polm.

Value

A (companion) matrix of dimensions \((mp,mp)\).

Examples

companion_matrix(polm(c(1,0,0,0.5,0))) # scalar polynomial
#>      [,1] [,2] [,3] [,4]
#> [1,]    0    0 -0.5    0
#> [2,]    1    0  0.0    0
#> [3,]    0    1  0.0    0
#> [4,]    0    0  1.0    0
companion_matrix(polm(diag(3)))        # zero degree polynomial 
#> <0 x 0 matrix>
companion_matrix(polm(dbind(d = 3, diag(2), -test_array(dim = c(2,2,1)))))
#>      [,1] [,2]
#> [1,]  111  121
#> [2,]  211  221
companion_matrix(polm(dbind(d = 3, diag(2), -test_array(dim = c(2,2,2)))))
#>      [,1] [,2] [,3] [,4]
#> [1,]  111  121  112  122
#> [2,]  211  221  212  222
#> [3,]    1    0    0    0
#> [4,]    0    1    0    0

if (FALSE) {
# the following examples throw an error
companion_matrix(polm(c(0,0,0,0.5))) # constant term is zero
companion_matrix(polm(test_array(dim = c(2,1,3)))) # non-square polynomial
companion_matrix(polm(test_array(dim = c(2,2,0)))) # zero polynomial
}