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

Abstract Decompositions of an orthogonal matrix $$\varvec{Q}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Q</mml:mi> </mml:mrow> </mml:math> are valuable on their own and play a crucial role in statistics by simplifying the often challenging estimation of $$\varvec{Q}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Q</mml:mi> </mml:mrow> </mml:math> 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 $$\varvec{Q}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Q</mml:mi> </mml:mrow> </mml:math> is sufficient; principal component analysis (PCA) is one such example. With this in mind, we propose a decomposition of $$\varvec{Q}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Q</mml:mi> </mml:mrow> </mml:math> , 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 $$\varvec{Q}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Q</mml:mi> </mml:mrow> </mml:math> 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 package provided as supplementary material, specifically designed to implement our decomposition. We illustrate its functionality using a benchmark dataset from the PCA literature.