Block Toeplitz matrix

Construct a block Toeplitz matrix from two 3-dimensional arrays R and C. The array R determines the first block row and C the first block column of the result. The \((i,j)\)-th block of the Toeplitz matrix is R[,,j-i+1] for \(j\geq i\) and C[,,i-j+1] for \(j < i\). In particular, note that the \((1,1)\) block is set to R[,,1] (while C[,,1] is ignored).

Usage

btoeplitz(R, C)

Arguments

R

Array with dimensions \((p,q,n)\). Corresponds to the first (block-) row, containing \(n\) matrices of dimension \((p \times q)\). A vector is coerced to an \((1,1,n)\) dimensional array and a \((p,q)\) matrix is interpreted as an \((p,q,1)\) dimensional array.

C

Array with dimensions \((p,q,m)\). Corresponds to the first (block-) column, containing \(m\) matrices of dimension \((p \times q)\). A vector is coerced to an \((1,1,m)\) dimensional array and a \((p,q)\) matrix is interpreted as an \((p,q,1)\) dimensional array.

Value

Matrix of size \(( pm \times qn )\).

Details

If only the argument R is provided then C is set to C = aperm(R,c(2,1,3))). So btoeplitz(R = X) is equivalent to btoeplitz(R = X, C = aperm(X,c(2,1,3))). Analogously btoeplitz(C = X) is equivalent to btoeplitz(R = aperm(X,c(2,1,3)), C = X). In these cases \(p=q\) must hold. The so constructed Toeplitz matrix is symmetric if and only if X[,,1] is symmetric. Note also that even in the case where only C is supplied the "R wins" rule holds, i.e. btoeplitz(C = X) sets the \((1,1)\) block equal to t(X[,,1]).

Examples

btoeplitz(0:3)
#>      [,1] [,2] [,3] [,4]
#> [1,]    0    1    2    3
#> [2,]    1    0    1    2
#> [3,]    2    1    0    1
#> [4,]    3    2    1    0
btoeplitz(0:3,-(0:3))
#>      [,1] [,2] [,3] [,4]
#> [1,]    0    1    2    3
#> [2,]   -1    0    1    2
#> [3,]   -2   -1    0    1
#> [4,]   -3   -2   -1    0

btoeplitz(R = test_array(dim=c(2,3,3)), C = -test_array(dim = c(2,3,2)))
#>      [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9]
#> [1,]  111  121  131  112  122  132  113  123  133
#> [2,]  211  221  231  212  222  232  213  223  233
#> [3,] -112 -122 -132  111  121  131  112  122  132
#> [4,] -212 -222 -232  211  221  231  212  222  232
btoeplitz(R = test_array(dim=c(2,2,1)), C = test_array(dim=c(2,2,0)))
#>      [,1] [,2]
btoeplitz(R = test_array(dim=c(2,2,3)))
#>      [,1] [,2] [,3] [,4] [,5] [,6]
#> [1,]  111  121  112  122  113  123
#> [2,]  211  221  212  222  213  223
#> [3,]  112  212  111  121  112  122
#> [4,]  122  222  211  221  212  222
#> [5,]  113  213  112  212  111  121
#> [6,]  123  223  122  222  211  221
btoeplitz(C = -test_array(dim=c(2,2,3)))
#>      [,1] [,2] [,3] [,4] [,5] [,6]
#> [1,] -111 -211 -112 -212 -113 -213
#> [2,] -121 -221 -122 -222 -123 -223
#> [3,] -112 -122 -111 -211 -112 -212
#> [4,] -212 -222 -121 -221 -122 -222
#> [5,] -113 -123 -112 -122 -111 -211
#> [6,] -213 -223 -212 -222 -121 -221
# create a symmetric matrix
X = test_array(dim= c(2,2,3))
X[,,1] = (X[,,1] + t(X[,,1]))/2
btoeplitz(R = X)
#>      [,1] [,2] [,3] [,4] [,5] [,6]
#> [1,]  111  166  112  122  113  123
#> [2,]  166  221  212  222  213  223
#> [3,]  112  212  111  166  112  122
#> [4,]  122  222  166  221  212  222
#> [5,]  113  213  112  212  111  166
#> [6,]  123  223  122  222  166  221
btoeplitz(C = X)
#>      [,1] [,2] [,3] [,4] [,5] [,6]
#> [1,]  111  166  112  212  113  213
#> [2,]  166  221  122  222  123  223
#> [3,]  112  122  111  166  112  212
#> [4,]  212  222  166  221  122  222
#> [5,]  113  123  112  122  111  166
#> [6,]  213  223  212  222  166  221

if (FALSE) {
# the following examples throw an error
btoeplitz(R = test_array(dim=c(2,1,3)), C = -test_array(dim = c(2,2,2)))
btoeplitz(R = test_array(dim=c(2,1,3)))
btoeplitz(C = test_array(dim=c(2,1,3)))
}