Decompositions of an orthogonal matrix \(\varvec{Q}\) are valuable on their own and play a crucial role in statistics by simplifying the often challenging estimation of \(\varvec{Q}\) 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 \(\varvec{Q}\) is sufficient; principal component analysis (PCA) is one such example. With this in mind, we propose a decomposition of \(\varvec{Q}\) , 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 \(\varvec{Q}\) 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.
