matrixcalc (version 1.0-6)

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)

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

Arguments

n

a positive integer value for the order of underlying matrices

Author

Frederick Novomestky

Details

The function firsts constructs an identity matrix of order $\frac{1}{2}n\left(n+1\right)$. ${\mathbf{u}}_{i,j}$ 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

u <- u.vectors( 3 )