Question

(c) Let A and B are Hermitian matrices, show that AB + BA is Hermitian. ​

Answer 1

To demonstrate AB + BA is Hermitian when A and B are Hermitian matrices, we find the conjugate transpose and use properties of Hermitian matrices to show (AB + BA)* = BA + AB, which is the same as AB + BA, confirming that AB + BA is Hermitian.

Step-by-step explanation:

Firstly, we know that for Hermitian matrices, A* = A and B* = B. { the asterisk denotes the conjugate transpose of a matrix. }

Using these properties, let's find the conjugate transpose of AB + BA:

(AB + BA)* = (AB)* + (BA)*

(AB)* = B*A* = BA because A and B are Hermitian

(BA)* = A*B* = AB because A and B are Hermitian

Now, after taking the conjugate transpose, we have:

(AB + BA)* = BA + AB

Since matrix addition is commutative, BA + AB is the same as AB + BA, therefore:

(AB + BA)* = AB + BA

Hence, the matrix AB + BA is indeed Hermitian.