How to get the best-sized bounding box from the covariance matrix and middle position?

Question

Given the covariance matrix and the average position calculated from the set of two-dimensional points, is there a way to simply calculate the bounding box or the best fit approximation (accuracy is not so important in this case)? The border can be rotated, and the position of each point is unknown. Could you help me?

Edited: I solved this simply by simply following a few equations here: http://www.visiondummy.com/2014/04/draw-error-ellipse-representing-covariance-matrix/

Answer 1

One thing you could try is to use the middle position as the center of your bounding box and rotate it to use your own covariance matrix vectors as your axes. See, for example, the diagram at http://en.wikipedia.org/wiki/Principal_component_analysis . This does not guarantee that you will receive the maximum possible bounding box - you can see this if you notice that all points, including inside the convex hull, that should not affect the minimum possible bounding box, will affect the eigenvectors - but it can be a decent approximation for some kinds of data.