LRDP: an R package implementing a new class of decompositions for orthogonal matrices

Decompositions of an orthogonal matrix Q are valuable on their own and play a crucial role in statistics by simplifying the often challenging estimation of Q when it is part of a model or method. It's important to note that, in some cases, any orthogonal matrix generated by permuting and/or flipping the signs of the columns of Q is sufficient; principal component analysis (PCA) is one such example. With this in mind, we propose a decomposition of Q, called LRDP, which allows control over the order and the sign of the columns. Due to its structure, our proposal enables the definition of simplified decompositions that can reproduce Q up to a permutation of the columns (LRD decomposition), up to a sign flip of the columns (LRP decomposition), or up to both (LR decomposition). Additionally, we introduce LRDP, an R package provided as supplementary material, specifically designed to implement our decomposition. We illustrate its functionality using a benchmark dataset from the PCA literature.