On Kronecker Products, Tensor Products And Matrix Differential Calculus
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.
|Date of creation:||Jul 2011|
|Date of revision:||Jul 2011|
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- 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.
- 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.
- Magnus, Jan R., 2010. "On the concept of matrix derivative," Journal of Multivariate Analysis, Elsevier, vol. 101(9), pages 2200-2206, October.
- Abadir,Karim M. & Magnus,Jan R., 2005. "Matrix Algebra," Cambridge Books, Cambridge University Press, number 9780521537469, December.
- Turkington,Darrell A., 2002. "Matrix Calculus and Zero-One Matrices," Cambridge Books, Cambridge University Press, number 9780521807883, December.
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