Many statistical problems involve the estimation of a \(\left( d\times d\right) \) orthogonal matrix \(\varvec{Q}\). Such an estimation is often challenging due to the orthonormality constraints on \(\varvec{Q}\). To cope with this problem, we use the well-known PLU decomposition, which factorizes any invertible \(\left( d\times d\right) \) matrix as the product of a \(\left( d\times d\right) \) permutation matrix \(\varvec{P}\), a \(\left( d\times d\right) \) unit lower triangular matrix \(\varvec{L}\), and a \(\left( d\times d\right) \) upper triangular matrix \(\varvec{U}\). Thanks to the QR decomposition, we find the formulation of \(\varvec{U}\) when the PLU decomposition is applied to \(\varvec{Q}\). We call the result as PLR decomposition; it produces a one-to-one correspondence between \(\varvec{Q}\) and the \(d\left( d-1\right) /2\) entries below the diagonal of \(\varvec{L}\), which are advantageously unconstrained real values. Thus, 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 \(\varvec{L}\). 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 a multivariate leptokurtic-normal distribution is assumed in each group. Compared to the commonly used normal distribution, the leptokurtic-normal has an additional parameter governing the excess kurtosis; this makes the estimation of \(\varvec{Q}\) in CPCA more robust against mild outliers. The usefulness of the PLR decomposition in leptokurtic-normal CPCA is illustrated by two biometric data analyses.

许多统计问题涉及\（\ left（d \ times d \ right）\）正交矩阵\（\ varvec {Q} \）的估计。由于\（\ varvec {Q} \）的正交性约束，这种估计通常具有挑战性。为了解决这个问题，我们使用公知的PLU分解，这因子分解任何可逆\（\左（d \倍Ð\右）\）矩阵的产物\（\左（d \倍ð\右）\）置换矩阵\（\ varvec {P} \），一个\（\ left（d \ times d \ right）\）单位下三角矩阵\（\ varvec {L} \）和一个\（\ left （d \ times d \ right）\）上三角矩阵\（\ varvec {U} \）。由于QR分解，当PLU分解应用于\（\ varvec {Q} \）时，我们找到了\（\ varvec {U } \）的形式。我们称结果为PLR分解。它在\（\ varvec {Q} \）与\（\ varvec {L} \对角线下方的\（d \ left（d-1 \ right）/ 2 \）条目之间产生一一对应的关系），这是不受限制的实数值。因此，一旦应用了分解，无论考虑中的目标函数如何，我们都可以使用任何经典的无约束优化方法来找到目标函数相对于\（\ varvec {L} \）的最小值（或最大值）。。出于说明目的，当在每组中假设多元瘦角正态分布时，我们在公共主成分分析（CPCA）中应用PLR分解来估计公共正交矩阵的最大似然。与通常使用的正态分布相比，瘦腿正态分布具有控制过度峰度的附加参数。这使得CPCA中对\（\ varvec {Q} \）的估计对于轻微的离群值更加稳健。两次生物特征数据分析说明了脂蛋白-正常CPCA中PLR分解的有用性。