Matrix factorization allows us to break down a complex matrix into a product of two or more simpler matrices. This is done with the intent of simplifying certain matrix operations, which can be more easily carried out on the factored matrices than on the original matrix. There are many different types of matrix factorizations, each with distinct properties and implications in practice. In this master's thesis, we will introduce five of the most common and fundamental matrix factorizations and their properties. We will demonstrate how using various matrix factorizations can solve systems of linear equations and compare their efficiency with the standard Gaussian elimination method. Additionally, we will introduce the linear least squares problem, where we will show, how to find the solution of overdetermined systems of linear equations using matrix factorizations. In both cases, we will compare the efficiency of using each type of factorization depending on the given system of equations. In the final part, we will present in a straightforward manner the application of each type of factorization across various scientific fields and with that, demonstrating the broad applicability of matrix factorizations.
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