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Many statistical problems involve the estimation of a $\left(d\timesd\right)$ orthogonal matrix $\textbf{Q}$. Such an estimation is oftenchallenging due to the orthonormality constraints on $\textbf{Q}$. To cope withthis problem, we propose a very simple decomposition for orthogonal matriceswhich we abbreviate as PLR decomposition. It produces a one-to-onecorrespondence between $\textbf{Q}$ and a $\left(d\times d\right)$ unit lowertriangular matrix $\textbf{L}$ whose $d\left(d-1\right)/2$ entries below thediagonal are unconstrained real values. Once the decomposition is applied,regardless of the objective function under consideration, we can use anyclassical unconstrained optimization method to find the minimum (or maximum) ofthe objective function with respect to $\textbf{L}$. For illustrative purposes,we apply the PLR decomposition in common principle components analysis (CPCA)for the maximum likelihood estimation of the common orthogonal matrix when amultivariate leptokurtic-normal distribution is assumed in each group. Comparedto the commonly used normal distribution, the leptokurtic-normal has anadditional parameter governing the excess kurtosis; this makes the estimationof $\textbf{Q}$ in CPCA more robust against mild outliers. The usefulness ofthe PLR decomposition in leptokurtic-normal CPCA is illustrated by twobiometric data analyses.

Unconstrained representation of orthogonal matrices with application to common principle components