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The conditions of a classification problem are as follows:

There are $K$ classes we are predicting with the $p$ dimensional predictor vector $X$. Let $Y$ be the $K-1$-dimensional one-hot encoded vector, with the $K^{th}$ class is encoded as $Y=(0, \ldots, 0)$.

If $S_{11}, S_{22}, S_{12}$ denote the sample covariance matrices (scaled by $N$ ) for $X, Y$, and $(X, Y)$, respectively, and if $S_{B}$ is the between-class covariance matrix, I want to show that $$ S_{B}=N S_{12} S_{22}^{-1} S_{21}. $$

Direct computations are welcome, but I would also like some intuition about what the between-class covariance matrix is.

Suzee
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  • In https://stats.stackexchange.com/a/237811/3277, section "Bonus instructions", a good (fast) way to compute B (between-class scatter matrix) is shown. (a covariance matrix, as you probably know, is a scatter matrix without division by "n", "n-1" or whatever its denominator.) – ttnphns Dec 08 '21 at 15:32

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