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We present a simple proof of the factorization of (complex) symmetric matrices into a product of a square matrix and its transpose, and discuss its application in establishing a uniqueness property of certain antilinear operators.
The $G^{+}$ method is a new method, a powerful one, for the study of (homogeneous and nonhomogeneous) products of nonnegative matrices -- for problems on the products of nonnegative matrices. To study such products, new classes of matrices…
It has been accepted that the polarization of the photon in vector beams is entangled with its momentum. Here a quantum description is advanced for the polarization that shows entanglement with the momentum. This is done by showing that the…
A classroom activity for high school physics students is presented to explore the distinction between correlation and causation. Using data linking ice cream sales to drowning deaths, presented within a fictional news article, students…
The purpose of this paper (which, in many passages, I take the liberty of writing in the first person singular for bringing a personal experience) is to show how teachers of Physics and Mathematics in Basic Education can teach, through a…
Physics makes powerful use of mathematics, yet the way this use is made is often poorly understood. Professionals closely integrate their mathematical symbology with physical meaning, resulting in a powerful and productive structure. But…
Mathematical understanding is built in many ways. Among these, illustration has been a companion and tool for research for as long as research has taken place. We use the term illustration to encompass any way one might bring a mathematical…
A higher dimensional generalization of the cross product is associated with an adequate matrix multiplication. This index-free view allows for a better understanding of the underlying algebraic structures, among which are generalizations of…
Ascertaining the physical state of a system is vital in order to understand and predict its behaviour. Tomography is a standard approach used to determine the form of an unknown state. Here we show that an alternative approach, based on…
Given two elements $a$ and $b$ of a noncommutative ring, we express $\left( ba\right)^n$ as a "row vector times matrix times column vector" product, where the matrix is the $n$-th power of a matrix with entries…
Transfer matrices and matrix product operators play an ubiquitous role in the field of many body physics. This paper gives an ideosyncratic overview of applications, exact results and computational aspects of diagonalizing transfer matrices…
Motivated by a recent finding that Fresnel's phenomenological description of the optical activity in the chiral medium is not self-consistent, we conduct a thorough investigation into the nature of the polarization of a plane light wave. We…
Many students in upper-division physics courses struggle with the mathematically sophisticated tools and techniques that are required for advanced physics content. We have developed an analytical framework to assist instructors and…
Making meaning with math in physics requires blending physical conceptual knowledge with mathematical symbology. Students in introductory physics classes often struggle with this, but it is an essential component of learning how to think…
This article introduces a pedagogical method for {\it solving combinatorial problems} that frequently involve structures that are unfamiliar or less familiar. Indeed, an indirect method has been proposed in order to evade any possible…
For several years, students visit us on different occasions at the university. But how to bridge from the school curriculum to the contents of the university mathematics? And how to find a focal point at which an active contribute, despite…
In the context of applying the Lorentz group theory to polarization optics in the frames of Stokes-Mueller formalism, some properties of the Lorentz group are investigated. We start with the factorized form of arbitrary Lorentz matrix as a…
In this paper, we study the recently defined notion of the inverse along an element. An existence criterion for the inverse along a product is given in a ring. As applications, we present the equivalent conditions for the existence and…
We use algebraic geometry to study matrix rigidity, and more generally, the complexity of computing a matrix-vector product, continuing a study initiated by Kumar, et. al. We (i) exhibit many non-obvious equations testing for (border)…
Courses in mathematical methods for physics students are not known for including too much in the way of mathematical rigour and, in some ways, understandably so. However, the conditions under which some quite commonly used mathematical…