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The commutation matrix: some theorems and applications

Author

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  • Magnus, Jan
  • Neudecker, H

Abstract

Tracy and Dwyer (1969) introduced a matrix that was later given the name 'permuted idertity matrix' by MacRae (1974). She defined it to be an (mn,mn) matrix partitioned into m by n .th th submatrices such that the i] submatrik has a 1 in its ji position and zeroes elsewhere. She showed that this matrix could be used for reversing the order of a Kronecker product, a property very useful in the calculation of matrix derivatives. In this paper it will therefore be called 'commutation matrix'. This commutation matrix will be applied to some problems in statistics. Expressions will be derived for the expectations and covariance matrices of stochastic vectors x 0 y, where x N n (p 1 ,V 1 ) and y N 2' V,) are stochastically dependent; m .! x 0 x, where x Nn(11,V). These results wi.4.1 then be applied to the central and noncentral Wishart distributions for which the variance matrices will be der:7ved. Various related results will also be reported.

Suggested Citation

  • Magnus, Jan & Neudecker, H, 1977. "The commutation matrix: some theorems and applications," University of Amsterdam, Actuarial Science and Econometrics Archive 293044, University of Amsterdam, Faculty of Economics and Business.
    • Handle: RePEc:ags:amstas:293044
    DOI: 10.22004/ag.econ.293044
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    Keywords

    Research Methods/ Statistical Methods;

    Statistics

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