For a standard tool in econometrics, matrix calculus, an approach is illustrated in this note that is unusual in that context, a coordinate-free approach. It can help to eliminate the persistent use of non-standard conventions. The Kronecker product and its use can be better understood. The complications and pitfalls of defining twice differentiability by partial derivatives are avoided. Its use is demonstrated, for example by giving a coordinate-free determination of the derivative of the determinant.

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Keywords Coordinate-free, Derivative of the determinant, Kronecker products, Matrix calculus, Maximum likelihood estimators
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Journal Journal of Multivariate Analysis
Brinkhuis, J. (2015). On the use of coordinate-free matrix calculus. Journal of Multivariate Analysis, 133, 377–381. doi:10.1016/j.jmva.2014.09.019