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.
u.vectors(n)A list with two named components
order \(n\) square matrix that maps each ordered pair (i,j) to a column in the identity matrix
order \(\frac{1}{2}n\left( {n + 1} \right)\) identity matrix
a positive integer value for the order of underlying matrices
Frederick Novomestky fnovomes@poly.edu
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 \(k = \left( {j - 1} \right)n + i - \frac{1}{2}j\left( {j - 1} \right)\).
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.