Linear Transformations

Scaling is a particular example of a linear transformation. Geometrically, scaling stretches or shrinks the axes in feature space. Clusters of data points that were originally spherical get stretched into ellipsoids, where the principal axes of these ellipsoids are aligned with the coordinate axes.

A more general linear transformation rotates as well as stretches the coordinates. Clusters of data points that were originally spherical get transformed into ellipsoids whose axes are rotated relative to the coordinate axes. This introduces a covariance between the components of the feature vector.

It is hard to imagine why we would purposely want to convert spherical clusters into ellipsoidal clusters. However, we very well might want to convert ellipsoidal clusters into spherical clusters. To do this, we need to understand more about the covariance.

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