Question

PAGE NO.
If A and B are Hermitian matrices, then show that AB + BA
is Hermitian and AB-BA is a skew Hermitian matrix​

Answer 1

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Answer 2

Answer:

Let A and B are Hermitian matrices then AB+BA is Hermitian and AB-BA is aloso Hermitian matrix.

Step-by-step explanation:

Step : 1

Since, A and B are Hermitian so that

A=A* and B=B*

Step : 2

To show that AB+BA is Hermitian

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

= B*A* + A*B*

=BA+AB

(AB+BA)*= AB+BA

Hence, AB+BA is Hermitian matrix.

Step : 3

To show that AB-BA is Hermitian matrix

(AB-BA)*=(AB)* - (BA)*

=B*A* - B*A*

=BA - AB

= - (-BA + AB)

= -(AB-BA)

Hence, AB-BA is Skew Hermitian matrix.