The Procrustes problem is a matrix optimization problem in which we aim to fit one data set to another as closely as possible. Given matrices $A$ and $B$ in their basic form, we seek an orthogonal matrix $Q$ such that the matrix $AQ$ is as close as possible to the matrix $B$. The difference between the matrices is measured using the Frobenius norm. In this thesis, we first consider the orthogonal Procrustes problem and its simplest generalization, the constrained orthogonal Procrustes problem, and derive the solution for both. We first derive the solution using the singular value decomposition, which provides a simple and numerically stable solution. We also present the method of Lagrange multipliers, which further explains the theoretical background of the problem. We then consider several important generalizations of the Procrustes problem and examples of its applications.
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