0
$\begingroup$

Crossposted from math.SE

Let $\mathbb P_n$ be the space of all $n \times n$ self-adjoint positive definite matrices. Consider the function $\varphi: \mathbb P_n \longrightarrow \mathbb R$ defined by $$\varphi (A) = -\text {tr}\ (A \log A).$$ Show that for all $t \in (0,1)$ $$\varphi ((1 - t) A + t B) \leq (1 - t) \varphi (A) + t \varphi (B) - \eta (t,1-t)$$ where $\eta (t,1 - t) = t \log (t) + (1 - t) \log (1 - t).$

I know that $\varphi$ is operator concave. But I don't have any idea as to how to bound $\varphi$ from above. Could anyone please give me some hint?

Thanks a bunch!

Improve this question
$\endgroup$
2
  • $\begingroup$ This is a website for professional mathematicians to ask each other questions about their research. Homework problems are off-topic here. $\endgroup$ Commented Aug 30, 2021 at 4:22
  • $\begingroup$ @Nik Weaver this is not a homework problem. I have found it in the exercise of my book in matrix analysis. Could you please give me some hint? I have also asked it in MSE but couldn't get any response from anybody. So I thought that it could very well be related to research problem in matrix analysis. That's why I have posted it here. Thanks. $\endgroup$ Commented Aug 30, 2021 at 4:40

0

Reset to default

You must log in to answer this question.

Start asking to get answers

Find the answer to your question by asking.

Ask question

Explore related questions

See similar questions with these tags.