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

Exercise 4

(i) Use the properties of trace to argue that tr(A’A) = tr(AA?) for any n x m matrix A.

(ii) For , verify that tr(A?A) = tr(AA?).

Step 1 of 2

If there are two matrixes with number of row and columns, let us say 2 rows and 2 columns, then the trace of multiplication of matrix with its transpose would be same irrespective of place of the matrix.

(i) This can be proved with the help of an example:

Trace (tr) of a matrix is nothing but the sum of its diagonal elements. The sum the diagonal elements of AA^” and A^”A is the same. It is. Therefore the following holds true: