On the use of coordinate-free matrix calculus

•A vector-space calculation of the derivative of the determinant function.•All conventions of matrix calculus arise naturally in the vector-space approach.•The vector-space approach clarifies the role of the Kronecker product in matrix calculus.•The vector-space approach gives a quick access to the vv-space in interior point methods.

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.