Rephrasing the Lengthy and Involved Proof of Kristof’s Theorem: A Tutorial with Some New Findings

Kristof’s theorem gives the global maximum and minimum of the trace of some matrix products without using calculus or Lagrange multipliers with various applications in psychometrics and multivariate analysis. However, the underutilization has been seen irrespective of its great use in practice. This may partially be due to the lengthy and involved proof of the theorem. In this tutorial, some known or new lemmas are rephrased or provided to understand the essential points in the proof. ten Berge’s generalized Kristof theorem is also addressed. Then, the modified Kristof and ten Berge theorems using parent orthonormal matrices are shown, which may be of use to see the properties of the Kristof and ten Berge theorems.