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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. – mining Jan 24 '12 at 21:38

For $n\geq 2$, the exponential is not convex, as the example $A=\pmatrix{1.1&1\\\ 1&1}$, $B=\pmatrix{1.1&0\\\ 0&1}$ shows. The computations are ugly, and done with sage. This can we extended to greater dimensions by completing by $0$ these matrices.

For the convexity of maps $X^p$, a good reference is Bathia's book Matrix Analysis. In problem V.5.4., we have to show that if $r<-1$ of $r>2$ this function is not convex.

I will add further details.

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