matrixcalc (version 1.0-3)

H.matrices: List of H Matrices

Description

This function constructs and returns a list of lists. The component of each sublist is derived from column vectors in an order r and order c identity matrix.

Usage

H.matrices(r, c = r)

Arguments

r

a positive integer value for an order r identity matrix

c

a positive integer value for an order c identify matrix

Value

A list with \(r\) components

1

A sublist of \(c\) components

2

A sublist of \(c\) components

...

r

A sublist of c components

Each component j of sublist i is a matrix {\bf{H}}_{i,j}

Details

Let \({{\bf{I}}_r} = \left[ {\begin{array}{*{20}{c}} {{{\bf{a}}_1}}&{{{\bf{a}}_2}}& \cdots &{{{\bf{a}}_r}} \end{array}} \right]\) be the order \(r\) identity matrix with corresponding unit vectors \({{{\bf{a}}_i}}\) with one in its \(i\)th position and zeros elsewhere. Let \({{\bf{I}}_c} = \left[ {\begin{array}{*{20}{c}} {{{\bf{b}}_1}}&{{{\bf{b}}_2}}& \cdots &{{{\bf{b}}_c}} \end{array}} \right]\) be the order \(c\) identity matrix with corresponding unit vectors \({{{\bf{b}}_i}}\) with one in its \(i\)th position and zeros elsewhere. The \(r \times c\) matrix \({\bf{H}}{}_{i,j} = {{\bf{a}}_i}\;{{\bf{b'}}_j}\) is used in the computation of the commutation matrix.

References

Magnus, J. R. and H. Neudecker (1979). The commutation matrix: some properties and applications, The Annals of Statistics, 7(2), 381-394.

Magnus, J. R. and H. Neudecker (1980). The elimination matrix, some lemmas and applications, SIAM Journal on Algebraic Discrete Methods, 1(4), December 1980, 422-449.

Examples

# NOT RUN {
H.2.3 <- H.matrices( 2, 3 )
H.3 <- H.matrices( 3 )
# }