# Kronecker Products and Matrix Calculus with Application — A. Graham (1981)

The 2018 Dover edition of this book is a compact little reference regarding various matrix theoretic notions not typically covered in a linear algebra class. However, I cannot recommend this for someone without a thorough background in matrix theory. The author presumes much familiarity, despite introducing basic topics like Kronecker products. Derivations and examples are difficult to follow due to constant back-references in an attempt to write as little as possible. Even for a reference, this would be difficult, as many of the useful tricks in matrix calculus are buried in examples, and would be difficult to locate using an index or table of contents. Thankfully, the book is short enough that one could just flip through.

Applications are not well-motivated, and are just a cursory mention without much context so the author can claim “applications” are discussed. The applications include solutions of special linear systems, a little control theory, and the general least squares problem in statistics. The author discusses “the permutation matrix”, when we obviously have more than one, but he refers to the permutation matrix that gives a relation between a matrix $X$ and $vec(X^{T})$. Much of the notation is lazy, and it takes a good deal of effort on the reader’s part to dissect exactly what we’re summing over, or what the dimensions of a particular elementary matrix are.

Overall, this book may serve as a compact reference, and it does attempt to reach an underserved readership interested in actual matrix calculus, but it misses its mark.

## Prerequisites

linear algebra, matrix theory, calculus

## Topics

• Matrix decompositions
• trace
• Vec operator
• Kronecker Products and applications to linear systems and basic control theory
• Matrix calculus (derivatives of vectors, chain rule, derivatives of scalar functions of a matrix with respect to a matrix)
• Matrix Differentials
• Derivatives of a matrix with respect to a matrix
• Applications of matrix calculus to least squares and maximum likelihood estimation of the multivariate normal parameters

### Attributes

 Difficulty 4 Good for teaching? 1 Proof Quality 2 Quality of Exercises 2 Self-Study 2 Readability 2