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In linear discriminant analysis (LDA), when there are fewer data instances than the number of dimensions (i.e., when the data matrix is of order $n \times m$ where $n$ is less than $m$), the within-class scatter matrix $S_W$ is said to be singular. Does this mean that a scatter matrix generated from any $n\times m$ data matrix where $n$ is less than $m$ is guaranteed to give a singular $S_W$ matrix? How much smaller does $n$ have to be in comparison to $m$ for $S_W$ matrix to be guaranteed singular?

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  • $\begingroup$ Yes, it means exactly that. $\endgroup$ – amoeba Sep 1 '15 at 23:08

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