Tensor products and matrix differential calculus
type: Item
instance of: scholarly article
title: Tensor products and matrix differential calculus
zbMATH Open document ID: 0576.15020
DOI: 10.1016/0024-3795(85)90194-6
published in: Linear Algebra and its Applications
publication date: 1985-01-01T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 15A69
Mathematics Subject Classification ID: 26B12
Mathematics Subject Classification ID: 15A04
zbMATH DE Number: 3920695
zbMATH Keywords: matrix derivative
zbMATH Keywords: vectorization
zbMATH Keywords: matrix equation
zbMATH Keywords: Kronecker products
zbMATH Keywords: tensor products
author: D. Stephen G. Pollock
MaRDI profile type: Publication
full work available at URL: 0024-3795(85)90194-6
OpenAlex ID: W1977074323
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description: scientific article; zbMATH DE number 3920695