01386nam a2200145 a 450000500170000000800400001710000170005724501040007426400160017833600090019433800200020352009050022350600460112885600660117420260901030417.0250101s2019    xx     o     000 0 eng d1 aLuca Bagnato10aUnconstrained representation of orthogonal matrices with application to common principle components 1barXivc2019  atext  aonline resource  aMany statistical problems involve the estimation of a $\left(d\times d\right)$ orthogonal matrix $\textbf{Q}$. Such an estimation is often challenging due to the orthonormality constraints on $\textbf{Q}$. To cope with this problem, we propose a very simple decomposition for orthogonal matrices which we abbreviate as PLR decomposition. It produces a one-to-one correspondence between $\textbf{Q}$ and a $\left(d\times d\right)$ unit lower triangular matrix $\textbf{L}$ whose $d\left(d-1\right)/2$ entries below the diagonal are unconstrained real values. Once the decomposition is applied, regardless of the objective function under consideration, we can use any classical unconstrained optimization method to find the minimum (or maximum) of the objective function with respect to $\textbf{L}$. For illustrative purposes, we apply the PLR decomposition in common principle components analysis (CP0 aOpen access — freely available to read.40uhttps://arxiv.org/pdf/1906.00587v1yRead the full paper (PDF)