matrixcalc (version 1.0-3)

E.matrices: List of E Matrices

Description

This function constructs and returns a list of lists. The component of each sublist is a square matrix derived from the column vectors of an order n identity matrix.

Usage

E.matrices(n)

Arguments

n

a positive integer for the order of the identity matrix

Value

A list with \(n\) components

1

A sublist of \(n\) components

2

A sublist of \(n\) components

...

n

A sublist of \(n\) components

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

Details

Let \({{\bf{I}}_n} = \left[ {\begin{array}{*{20}{c}} {{{\bf{e}}_1}}&{{{\bf{e}}_2}}& \cdots &{{{\bf{e}}_n}} \end{array}} \right]\) be the order \(n\) identity matrix with corresponding unit vectors \({{{\bf{e}}_i}}\) with one in its \(i\)th position and zeros elsewhere. The \(n \times n\) matrix \({{\bf{E}}_{i,j}}\) is computed from the unit vectors \({{{\bf{e}}_i}}\) and \({{{\bf{e}}_j}}\) as \({{\bf{E}}_{i,j}} = {{\bf{e}}_i}\;{{\bf{e'}}_j}\). These matrices are stored as components in a list of lists.

References

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.

Magnus, J. R. and H. Neudecker (1999). Matrix Differential Calculus with Applications in Statistics and Econometrics, Second Edition, John Wiley.

Examples

# NOT RUN {
E <- E.matrices( 3 )
# }