Applications of the diagonal representation II
4. Square root of a matrix
Definition 1. For a symmetric matrix with non-negative eigenvalues the square root is defined by
Exercise 1. (1) is symmetric and satisfies
Proof. By properties of orthogonal matrices
5. Generalized least squares estimator
The error term in the multiple regression under homoscedasticity and in absence of autocorrelation satisfies
(2) where is some positive number.
The OLS estimator in this situation is given by
Now consider a more general case
Exercise 2. The variance matrix is always symmetric and non-negative.
Exercise 3. Let's assume that is positive. Show that is symmetric and satisfies
Proof. By Exercise 1 the eigenvalues of are positive. Hence its inverse exists and is given by where It is symmetric as an inverse of a symmetric matrix. It remains to apply Exercise 1 to
Exercise 4. Find the variance of .
Solution. Using the definition of variance of a vector
Exercise 4 suggests how to transform to satisfy (2). In the equation
the error satisfies the assumption under which (2) is applicable. Let Then we have and from (3) Since this can be written as