Introductory Econometrics: A Modern Approach 6th Edition by Jeffrey M Wooldridge

Exercise 6

(i) Show that if A is an n x n symmetric, positive definite matrix, then A must have strictly positive diagonal elements.

(ii) Write down a 2 x 2 symmetric matrix with strictly positive diagonal elements that is not positive definite.

Step 1 of 2

A positive definite matrix should have the following two properties:

- A positive diagonal element

- The diagonal element should be greater than the sum of the row and column elements in which the positive diagonal lies.

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(i) Let us assume that A is a positive definite matrix with negative diagonal elements

This would clearly give a negative result if x>y. This means A has to be positive with its diagonal elements if A is positive definite matrix.