0
$\begingroup$

A given problem states:

If $A$ and $B$ are hermitian matrices, prove that $(AB+BA)$ is hermitian.

Because the sum of hermitian matrices is known to be hermitian, the problem seems to me to boil down to proving that the products $AB$ and $BA$ are hermitian. But this isn't necessarily so, unless $AB = BA$.

This problem is part of a course in quantum mechanics, and my attempt goes something like this: $$ \int\psi^*(AB+BA)\psi \mathop{dr} = \int\psi^*AB\psi \mathop{dr}+\int\psi^*BA\psi \mathop{dr}\\ $$ At this point I suspect one should use the properties of $A$ and $B$ to get to the answer $$ \int(AB+BA)^*\psi^*\psi \mathop{dr} $$

Any clarification would be appreciated!

$\endgroup$
1
$\begingroup$

$(AB+BA)^\ast = (AB)^\ast + (BA)^\ast = B^\ast A^\ast + A^\ast B^\ast = BA+AB = AB+BA$.

Individually $AB$ and $BA$ are not Hermitian, but $AB+BA$ is.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.