This paper considers the principal component analysis when the covariance matrix of the input vectors drops rank. This case sometimes happens when the total number of the input vectors is very limited. First, it is found that the eigen decomposition of the covariance matrix is not uniquely defined. This implies that different transform matrices could be obtained for performing the principal component analysis. Hence, the generalized form of the eigen decomposition of the covariance matrix is given. Also, it is found that the matrix with its columns being the eigenvectors of the covariance matrix is not necessary to be unitary. This implies that the transform for performing the principal component analysis may not be energy preserved. To address this issue, the necessary and sufficient condition for the matrix with its columns being the eigenvectors of the covariance matrix to be unitary is derived. Moreover, since the design of the unitary transform matrix for performing the principal component analysis is usually formulated as an optimization problem, the necessary and sufficient condition for the first order derivative of the Lagrange function to be equal to the zero vector is derived. In fact, the unitary matrix with its columns being the eigenvectors of the covariance matrix is only a particular case of the condition. Furthermore, the necessary and sufficient condition for the second order derivative of the Lagrange function to be a positive definite function is derived. It is found that the unitary matrix with its columns being the eigenvectors of the covariance matrix does not satisfy this condition. Computer numerical simulation results are given to valid the results.

中文翻译：

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使用降秩协方差矩阵进行主成分分析

本文考虑了输入向量协方差矩阵降秩时的主成分分析。当输入向量的总数非常有限时，有时会发生这种情况。首先，发现协方差矩阵的特征分解不是唯一定义的。这意味着可以获得不同的变换矩阵来执行主成分分析。因此，给出了协方差矩阵的特征分解的广义形式。此外，发现列是协方差矩阵的特征向量的矩阵不一定是酉矩阵。这意味着用于执行主成分分析的变换可能不会保留能量。为了解决这个问题，推导出其列为协方差矩阵的特征向量的矩阵是酉矩阵的充要条件。此外，由于用于执行主成分分析的酉变换矩阵的设计通常被表述为优化问题，因此导出了拉格朗日函数的一阶导数等于零向量的充分必要条件。事实上，其列是协方差矩阵的特征向量的酉矩阵只是该条件的一个特例。进一步推导出拉格朗日函数的二阶导数为正定函数的充要条件。发现列是协方差矩阵的特征向量的酉矩阵不满足这个条件。给出了计算机数值模拟结果来验证结果。