01432nam a2200169 a 4500 20260901030417.0 250101s2019 xx o 000 0 eng d Luca Bagnato Unconstrained representation of orthogonal matrices with application to common principle components arXiv 2019 text online resource Many 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 (CP Open access — freely available to read. https://arxiv.org/pdf/1906.00587v1 Read the full paper (PDF) ERES 717 717 0 0 0 3 GEN MAIN MAIN SCICOMP 2026-09-01 0 YGE000978 2026-09-01 03:04:18 https://arxiv.org/pdf/1906.00587v1 2026-09-01 PAPER