## 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 (2) Since this can be written as

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