Missing step(s) in a proof that includes inner product and matrices.

Question

How do I get from

$-\langle A^{*}Pe^{tA}x_0, \ e^{tA}x_0 \rangle-\langle PAe^{tA}x_0, \ e^{tA}x_0 \rangle$

to

$-\langle PAe^{tA}x_0, \ e^{tA}x_0 \rangle-\langle Pe^{tA}x_0, \ Ae^{tA}x_0 \rangle$?

Answer 1

The order of the terms is mixed up, which makes it more confusing than necessary. Look at the first term in the first equation and the second term in the second equation: Here, only the property of $A^*$ being the adjoint of $A$ is used. The other cross terms are identical.