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How to determine the convexity of the following functions:

  • $X^p$, in which $p$ is a real number and $X$ is $n \times n$ symmetric positive definite matrix.

  • $e^X$ in which $X$ is a $n \times n$ symmetric matrix and $n \geq 2$

The convexity mentioned above refers to the proper cone $S^n_{+}$.

Thanks a lot!

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    Yeah, you are right. The convexity of the first function depends on $p$. If $1 \leq p \leq 2$ or $-1 \leq p \leq 0$, $X^p$ is matrix convex. If $0 \leq p \leq 1$, $X^p$ is matrix concave. But I do not know how to deduce the result.2012-01-24

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