Kronecker Products and Matrix Calculus with Applications — A. Graham

This little book is quite dense and wastes no time on wordiness. Its compact nature packs a lot of useful matrix calculus into just over 100 pages with no frills, yet still manages to give a decent number of examples to illustrate concepts, including applications (most in the multivariate statistics realm). For engineers especially who need to learn how to do some matrix calculus, this is a good resource. Mathematicians interested in proofs and deeper notions may find it a bit lacking in the regard and should probably supplement with another text to pursue this subject more deeply. Overall, I found it definitely worth the read, and now have a great little resource for those times when I need to recall some tricks. Solutions to problems are given in the text, and the bibliography is short but good. 



calculus, matrix algebra, some linear algebra is helpful

Topics Covered

  • matrix decomposition
  • vec operator
  • Kronecker product and Kronecker sum (properties, rules, etc)
  • applications of the Kronecker product in solving linear systems and differential equations
  • derivatives of vectors
  • derivative of scalar functions with respect to a matrix
  • derivative of matrix with respect to one of its elements and conversely
  • derivatives of matrix powers
  • matrix differential
  • derivative of a matrix with respect to a matrix
  • applications
    • least squares
    • constrained optimization
    • maximum likelihood estimation of multivariate normal distribution
    • Jacobians of transformations


Difficulty 2
Good for teaching? 4
Proof Quality 2
Quality of Exercises 4
Self-Study 3
Readability 4