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What is an intuitive way to think about the covariance matrix in an N≥3 dimensional Normal distribution?

In two dimensions the covariance matrix can be visualized by plotting a region of constant probability (below.) This is an ellipse and the covariance matrix determines its rotation. But is there a three-dimensional analogy to this view, or else what is a good way to understand covariance in higher dimensional Normal distributions?

2D Normal distribution joint probability

Luke Gorrie
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    It looks like a (American) football. – Dave Oct 12 '21 at 09:48
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    Suppose you want to switch it from an American football to an upright Rugby ball or a Soccer ball. How do you manipulate the covariance matrix to achieve this? – Luke Gorrie Oct 12 '21 at 11:14
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    One way is to apply the 2D explanation at https://stats.stackexchange.com/a/62147/919 in multiple dimensions: no changes need to be made; everything carries through. A slightly different insight is afforded by [contemplating how to create an arbitrary covariance matrix](https://stats.stackexchange.com/questions/215497). Normal distributions enjoy a close relationship between covariance matrices and multiple least squares regression. For a 2D account see https://stats.stackexchange.com/a/71303/919. It also generalizes to higher dimensions (relating parabolic subgroups to rotational subgroups). – whuber Oct 12 '21 at 15:16
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    Mathematically it could be imagined as ellipsoid, https://stats.stackexchange.com/questions/164741/how-to-find-the-maximum-axis-of-ellipsoid-given-the-covariance-matrix – msuzen Oct 12 '21 at 16:32

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