A Review of Matrix Algebra for Power and Energy Applications

TL;DR

The paper surveys essential matrix algebra techniques and their Julia implementations for power and energy applications, highlighting practical visualization, data handling, and code organization. It covers vectors, matrices, and hyper-arrays; linear transformations and eigenanalysis; quadratic forms and norms; linear systems and sparse representations; and data management with DataFrames and CSV, all within the context of power-system modeling such as convergence of power flows. The work emphasizes computational efficiency (e.g., LU factorization, Cholesky, pseudoinverse for singular systems, sparse matrices) and software engineering practices to produce modular, scalable Julia scripts. Together, these elements enable robust mathematical analysis and efficient simulation in modern power systems.

Abstract

This report presents a brief review of matrix algebra and its implementation in Julia for power and energy applications. First, we present basic examples of data visualization, followed by conventional operations with matrices and vectors. Then, we study quadratic forms and norms, two main concepts required in the convergence study of the power flow in power and energy applications. After that, we give good practices to create a neat code in Julia. There is an extensive set of examples available on the Internet related to these basic aspects, so we avoid repeating what is well documented. Hence, we show only basic examples to create our first scripts.

Figures (5)

• Figure 1: Example of a vector, a matrix, and a three-dimensional hyper-array
• Figure 2: Example of a parallelogram ($\mathbb{R}^2$) or a parallepiped ($\mathbb{R}^3$) formed by a set of linearly independent vectors.
• Figure 3: Example of a linear transformation. The eigenvectors $v$ represent the direction which remains after the transformation.
• Figure 4: Example of two different quadratic forms: left) $q(x_1,x_2)=x_1^2+3x_1x_2-x_2^2$, right) $q(x_1,x_2)=0.6x_1^2+0.4x_1x_2+0.8x_2^2$. The latter is a paraboloid defined by a quadratic form with a positive definite matrix.
• Figure 5: Visualization of a sparse matrix. Dots represent entries different from zero.