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!


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

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


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