Quadrics associated with symmetric matrices

1 Quadrics associated with symmetric matrices Given A x a symmetric matrix, the quadratic form S = xTAx can be rewritten as 

Using the factorization of the symmetric matrix A as U UT, we get, upon a change of coordinates 

In the previous two equations, the old coordinates x and y are set as linear combinations of the eigenvectors. The following equation is therefore arrived at 

The purpose of this exercise is to learn how to draw a probability ellipse from the mean values and covariance matrix, a topic to be further developed in Chapter 4. Na and Cl are incorporated into clouds during evaporation of seawater and are therefore strongly correlated. 

Let us call x the vector of Na and Cl concentrations, x the vector of sample means and S the symmetric, positive-definite covariance matrix, i.e., the 2 x 2 matrix with variances on the diagonal and the covariance between Na and Cl concentrations as off-diagonal terms. The equation of the ellipse to be drawn can be written 

---