# Expected Value of a Polar Decomposition

What is a method of computing quantities such as $\mathbb{E} [ \text{tr} (NU \mathcal{P} (M U)^t)]$ where $\mathcal{P}$ is the polar decomposition, $M,N \in \mathbb{R}^{n \times n}$, and $U$ is a random $n\times n$ orthogonal matrix over the Haar measure. In dimension 2, over SO(2), the quantity can be related to the product $MN^t$, but I would like to know if there is some lower bound in terms of this product in higher dimensions.

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