matrixcalc (version 1.0-3)

u.vectors: u vectors of an identity matrix

Description

This function constructs an order n * ( n + 1 ) / 2 identity matrix and an order matrix u that that maps the ordered pair of indices (i,j) i=j, ..., n; j=1, ..., n to a column in this identity matrix.

Usage

u.vectors(n)

Arguments

n

a positive integer value for the order of underlying matrices

Value

A list with two named components

k

order \(n\) square matrix that maps each ordered pair (i,j) to a column in the identity matrix

I

order \(\frac{1}{2}n\left( {n + 1} \right)\) identity matrix

Details

The function firsts constructs an identity matrix of order \(\frac{1}{2}n\left( {n + 1} \right)\). \({{{\bf{u}}_{i,j}}}\) is the column vector in the order \(\frac{1}{2}n\left( {n + 1} \right)\) identity matrix for column \(k = \left( {j - 1} \right)n + i - \frac{1}{2}j\left( {j - 1} \right)\).

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

Run this code
# NOT RUN {
u <- u.vectors( 3 )
# }

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