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
(1)
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
(3)
Now consider a more general case
Exercise 2. The variance matrix is always symmetric and non-negative.
Proof.
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