Tensor products and matrix differential calculus
First some of the fundamental results of the Kronecker products of matrices are expressed in the form of tensor products. Denote V and W two real coordinate vector spaces of dimensions n and m. For \(X=[x_{ij}]\) in \(V^*\otimes W\), a column-vectorization operator c is defined by \(X^ c=\sum_{ij}x_{ij}(e_ j\otimes e_ i)\), where \(e_ j\otimes e_ i\) is an mn\(\times 1\) column vector with a unit in the \([(j- 1)m+i]th\) position and zeros elsewhere. Then, a matrix equation of the form \(Y=AXB'\) can be written as a vector equation \(Y^ c=(B\otimes A)X^ c\). For a matrix function \(Y=Y(X)\) with respect to its matrix argument X, the author considers three alternative definitions of the matrix derivative of Y(X) and establishes relationships amongst them, and he shows that only one (vectorial definition) of three definitions, that is, \(\partial Y^ c/\partial X^ c=\sum \partial y_{k\ell}\partial x_{ij}(e_{\ell}\otimes e_ k\otimes e^{*j}\otimes e^{*i})\) in terms of the natural bases, is viable.
scientific article; zbMATH DE number 3920695
scientific article