4
$\begingroup$

For three symmetric positive-semidefinite matrices $A,B,C$ I am trying to figure out if the following inequality holds, at least in some cases:

$$tr(Ae^{B+C})≤tr(Ae^Be^C)$$ Note that if $A=I$ then this is the Golden Thompson inequality:

$$tr(e^{B+C})≤tr(e^Be^C)$$

I wonder if the first inequality is true or if adding certain assumptions (such as $A≼I$ i.e. $I−A$ is positive semidefinite) will suffice to prove it. I have tried searching for similar inequalities with no luck.

$\endgroup$
2
$\begingroup$

The difficulties with generalizations of the Golden-Thompson inequality to three matrices arise because the trace of a product of three positive symmetric matrices is in general not positive; unlike the trace of the product of two positive symmetric matrices, which is positive: ${\rm tr}\,e^B e^C={\rm tr}\,XX^t=\sum_{n,m}X_{nm}^2>0$, with $X=e^{B/2}e^{C/2}$ and $X^t$ the transpose of $X$.

So while ${\rm tr}\,Ae^{B+C}$ is a positive number, ${\rm tr}\,Ae^Be^C$ can be negative.

There do exist three-matrix generalizations of the Golden-Thompson inequality, but they take an entirely different form, see A survey of certain trace inequalities, equation 21.

$\endgroup$

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.