Econ4261_Hw1_key

# Econ4261_Hw1_key - ECON 4261 Introduction to Econometrics...

This preview shows pages 1–3. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: ECON 4261: Introduction to Econometrics Fall 2009 Problem Set 1 – Answer Key Exercise 1 Let A be an ( m × n ) matrix, with m > n . Show that the matrix A A is symmetric, and that if A has full rank, A A is positive definite. Let A = a 11 a 12 ... a 1 n a 21 a 22 ... a 2 n . . . . . . . . . . . . a m 1 a m 2 ... a mn . Then A A = a 11 a 21 ... a m 1 a 12 a 22 ... a mn . . . . . . . . . . . . a 1 n a 2 n ... a mn a 11 a 12 ... a 1 n a 21 a 22 ... a 2 n . . . . . . . . . . . . a m 1 a m 2 ... a mn = ∑ m j =1 a 2 j 1 ∑ m j =1 a j 1 a j 2 ... ∑ m j =1 a j 1 a jn ∑ m j =1 a j 2 a j 1 ∑ m j =1 a 2 j 2 ... ∑ m j =1 a j 2 a jn . . . . . . . . . . . . ∑ m j =1 a jn a j 1 ∑ m j =1 a jn a j 2 ... ∑ m j =1 a 2 jn and note that the elements off the diagonal of this matrix are the same. Hence A A is symmetric. To prove A A is positive definite, we must prove that q = x ( A A ) x > 0 for all x 6 = 0 . Suppose this does not happen, i.e. we have x ( A A ) x ≤ 0 for some x 6 = 0 . Note that q = x ( A A ) x = ( x A )( Ax ) = ( Ax ) ( Ax ) and Ax is an ( m × 1) vector. So ( Ax ) ( Ax ) is the inner product of the vector Ax with itself. Hence, it’s a sum of squares. So it cannot be the case that q < 0. Hence the only possibility is to have q = 0 . Since q is a sum of squares, the only possibility is that all the components of the sum are zero. This would imply that Ax = 0. But since we are assuming that A has full rank (and m > n ), we must have x = 0 . But this contradicts our initial hypothesis that x 6 = 0. Hence we must have q = x ( A A ) x > 0 for all x 6 = 0 . So A A is positive definite. 1 Exercise 2 Characterize each of the following statements as true or false. If a statement is true, show the proof. If it is false, give a counterexample. Assume that A and B are square matrices....
View Full Document

{[ snackBarMessage ]}

### Page1 / 10

Econ4261_Hw1_key - ECON 4261 Introduction to Econometrics...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online