Properties of a matrix-valued generalization of the $\Gamma$ function - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T10:20:56Z http://mathoverflow.net/feeds/question/78651 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/78651/properties-of-a-matrix-valued-generalization-of-the-gamma-function Properties of a matrix-valued generalization of the $\Gamma$ function Ralph Furmaniak 2011-10-20T07:41:08Z 2011-11-03T21:22:09Z <p>I am interested in the following function (Mellin transform of matrix exponential): $$\int_0^{\infty} x^{s-1} e^{-A-Bx}d x$$ Where $x$ and $s$ are scalars, but $A$ and $B$ are matrices with $B\succ 0$.</p> <p>When $A$ and $B$ commute then it is easy to compute: simultaneously diagonalizing gives gamma functions on the diagonal. Are there any other cases in which you can say something about this function? Has it been studied? Is there an analytic continuation?</p> http://mathoverflow.net/questions/78651/properties-of-a-matrix-valued-generalization-of-the-gamma-function/78699#78699 Answer by S. Sra for Properties of a matrix-valued generalization of the $\Gamma$ function S. Sra 2011-10-20T19:58:12Z 2011-11-03T21:22:09Z <p>This is not an answer, just an overgrown comment. I hope it suggests something non-boring.</p> <p>Assume that $-A$ is Hermitian, and $B \ge 0$. Let $G(s)$ denote your matrix valued integral.</p> <p>Then, we have (am writing $A$ instead of $-A$ to be consistent with some standard notation):</p> <p>$$g(s) := \mbox{tr}(G(s)) = \int_0^\infty x^{s-1}\mbox{tr}(e^{A-x B})dx.$$</p> <p>Assuming that the <a href="http://www.mat.univie.ac.at/~schachermayer/pubs/preprnts/prpr0120.pdf" rel="nofollow">BMV Conjecture</a> is true (see e.g., <a href="http://arxiv.org/abs/1107.4875" rel="nofollow">apparent proof</a>), we can represent the trace term as:</p> <p>$$\mbox{tr}(e^{A-x B}) = \int_0^\infty e^{-tx}d\mu(t; A,B),$$</p> <p>where $\mu(; A, B)$ is a positive measure supported by $[0,\infty)$.</p> <p>Then, assuming that the order of integration can be changed, we have</p> <p>$$g(s) = \int_0^\infty\int_0^\infty e^{-tx}x^{s-1}dxd\mu(t) = \int_0^\infty \Gamma(s)t^{-s}d\mu(t; A,B),$$ at which point I run out of ideas. </p> <p>PS: Do you care about just $2 \times 2$ matrices, or then can be $n \times n$?</p>