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On Kronecker Products, Tensor Products And Matrix Differential Calculus

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  • Stephen Pollock

Abstract

The algebra of the Kronecker products of matrices is recapitulated using a notation that reveals the tensor structures of the matrices. It is claimed that many of the difficulties that are encountered in working with the algebra can be alleviated by paying close attention to the indices that are concealed beneath the conventional matrix notation. The vectorisation operations and the commutation transformations that are common in multivariate statistical analysis alter the positional relationship of the matrix elements. These elements correspond to numbers that are liable to be stored in contiguous memory cells of a computer, which should remain undisturbed. It is suggested that, in the absence of an adequate index notation that enables the manipulations to be performed without disturbing the data, even the most clear-headed of computer programmers is liable to perform wholly unnecessary and time-wasting operations that shift data between memory cells.

Suggested Citation

  • Stephen Pollock, 2014. "On Kronecker Products, Tensor Products And Matrix Differential Calculus," Discussion Papers in Economics 14/02, Division of Economics, School of Business, University of Leicester.
    • Handle: RePEc:lec:leecon:14/02
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    File URL: https://www.le.ac.uk/economics/research/RePEc/lec/leecon/dp14-02.pdf
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    References listed on IDEAS

    as
    1. Abadir,Karim M. & Magnus,Jan R., 2005. "Matrix Algebra," Cambridge Books, Cambridge University Press, number 9780521537469, January.
    2. Turkington, Darrell, 2000. "Generalised vec operators and the seemingly unrelated regression equations model with vector correlated disturbances," Journal of Econometrics, Elsevier, vol. 99(2), pages 225-253, December.
    3. Turkington,Darrell A., 2002. "Matrix Calculus and Zero-One Matrices," Cambridge Books, Cambridge University Press, number 9780521807883, January.
    4. repec:cup:cbooks:9780521822893 is not listed on IDEAS
    5. Magnus, Jan R., 2010. "On the concept of matrix derivative," Journal of Multivariate Analysis, Elsevier, vol. 101(9), pages 2200-2206, October.
    6. Magnus, J.R. & Neudecker, H., 1979. "The commutation matrix : Some properties and applications," Other publications TiSEM d0b1e779-7795-4676-ac98-1, Tilburg University, School of Economics and Management.
    Full references (including those not matched with items on IDEAS)

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    Cited by:

    1. Mutschler, Willi, 2014. "Identification of DSGE Models - A Comparison of Methods and the Effect of Second Order Approximation," VfS Annual Conference 2014 (Hamburg): Evidence-based Economic Policy 100598, Verein für Socialpolitik / German Economic Association.
    2. Mutschler, Willi, 2015. "Identification of DSGE models—The effect of higher-order approximation and pruning," Journal of Economic Dynamics and Control, Elsevier, vol. 56(C), pages 34-54.

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    More about this item

    Keywords

    linear and multilinear algebra; matrix theory; tensor products; Kronecker products; index notation; matrix differential calculus;
    All these keywords.

    JEL classification:

    • C8 - Mathematical and Quantitative Methods - - Data Collection and Data Estimation Methodology; Computer Programs
    • C65 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Miscellaneous Mathematical Tools

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