| # Exercises Semi-Definite Programming > Exercises ## Optimization of ellipsoid shapes For , with symmetric and positive-definite, we define the ellipsoid A measure of the ‘‘size’’ of the ellipsoid is , with the trace (the sum of the diagonal elements of the matrix argument). Motivate our choice of the size function. Hint: Figure out the the semi-axis lengths of the ellipsoid as a function of . Show that , where is a factor of (any matrix such that ; in matlab, can be obtained via the command `chol`.) Show that for any , positive-definite, the set is contained in if and only if is positive semi-definite. For given symmetric matrices , , both positive-definite, show how to compute an ellipsoid, centered at the origin, that contains both and has minimum size. Implement and plot your result with the data contained in here. (The function for symmetric and positive definite, is implemented in CVX using `trace_inv`.)