Suppose I have a covariance matrix A = [5 0;0 9]; This matrix represent an ellipse, where the two variables x1 and y1 forming the ellipse has a variance of 5 and 9 respectively. Now, if I want to rotate this matrix by 45 degrees, we get

A2 = R*A = [0.707 0.707;-0.707 0.707] = [7 2;2 7]; The variance of rotated matrix A2 is outside the limit of A, that is, x2 = 7, y2= 7 for A2. Now I want to stretch A2 in such a way that x2