Page 116 - Matrix Analysis & Applied Linear Algebra

110 Chapter 3 Matrix Algebra

Proof. By definition,
T
(AB) =[AB] ji = A j∗ B ∗i .
ij
T
Consider the (i, j)-entry of the matrix B A T and write
T T T T T T
B A = B A = B A
ij i∗ ∗j ik kj
k

= [B] ki [A] jk = [A] jk [B] ki
k k
= A j∗ B ∗i .
T T T T T T
Therefore, (AB) = B A for all i and j, and thus (AB) = B A .
ij ij
The proof for the conjugate transpose case is similar.
Example 3.6.4
T
For every matrix A m×n , the products A A and AA T are symmetric matrices
because
T T T T T T T T T T T T
A A = A A = A A and AA = A A = AA .

Example 3.6.5

Trace of a Product. Recall from Example 3.3.1 that the trace of a square
matrix is the sum of its main diagonal entries. Although matrix multiplication
is not commutative, the trace function is one of the few cases where the order of
the matrices can be changed without affecting the results.
Problem: For matrices A m×n and B n×m , prove that
trace (AB)= trace (BA).

Solution:

trace (AB)= [AB] ii = A i∗ B ∗i = a ik b ki = b ki a ik
i i i k i k

= b ki a ik = B k∗ A ∗k = [BA] kk = trace (BA).
k i k k
Note: This is true in spite of the fact that AB is m × m while BA is n × n.
Furthermore, this result can be extended to say that any product of conformable
matrices can be permuted cyclically without altering the trace of the product.
For example,
trace (ABC)= trace (BCA)= trace (CAB).
However, a noncyclical permutation may not preserve the trace. For example,
trace (ABC)
= trace (BAC).