Non-commutative arithmetic-geometric mean inequalities.

The arithmetic-geometric mean inequality states that the arithmetic mean of a set of positive numbers is always greater than the geometric mean unless all of the numbers are equal to one another. Does a similar inequality hold for positive definite matrices? The answer is not straightforward as positive definite matrices do not commute and there are myriad possible connotations of “greater than.” I will discuss how we might use a “non-commutative” arithmetic geometric mean inequality in control, signal processing, and optimization applications, but will also show that such an inequality does not generally hold. Instead, I will describe a weaker inequality that is true, and explore some conjectures about how this bound might be strengthened.