Let $A,D\in \mathbb{C}^{n \times n}$ be diagonal matrices. I need to calculate $$\int_{U(n)}\det{(A-HDH^\dagger)}\,\mathrm{d}H$$ where $dH$ is the unit invariant Haar measure on the group of unitary matrices and $H^\dagger$ is the conjugate transpose of $H$. (If $A=I$ this is very easy to solve, but I want the answer for $A\neq I$ in terms of $A$ and $D$.)
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2$\begingroup$ First of all, the matrices should probably be in $\mathbb{C}^{n^2}$ not in $\mathbb{C}^n$. Second, what role does $D$ play? I don't see it in the integral, perhaps $D = L$? Third, what does $H'$ mean? Is it the transpose of $H$? Fourth, are the words "deterministic" really relevant here? $\endgroup$– Vít TučekCommented Jan 28, 2015 at 12:08
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$\begingroup$ Oops, Sorry for these typos! I edited them. Does it make sense now? Thanks. $\endgroup$– PeterCommented Jan 28, 2015 at 17:22
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1$\begingroup$ Wlog one can assume that $A$ is diagonal because the Haar measure is invariant under left- and rightmultiplication. $\endgroup$– Johannes HahnCommented Jan 28, 2015 at 17:28
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$\begingroup$ Do you need an exact answer or would large $n$ asymptotics suffice? In the second situation, in the case where $A$ and $D$ are diagonal with free entries, one might be able to use an approximation of the spectral distribution of $A-HDH^*$ ($H$ chosen uniformly at random) by the free convolution of $\mu_A$ with $\mu_D$, see e.g. the first few pages of arxiv.org/abs/math/9809193 $\endgroup$– Yemon ChoiCommented Jan 28, 2015 at 18:17
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1$\begingroup$ @Peter: Just a follow up on Carlo's suggestion. I don't think you want to compute the integral of exp sum of traces as in your last comment. $\int\exp(t\ {\rm tr}(A^{-1}HDH^{\dagger}))dH$ is enough, then take derivatives in $t$. That's the famous Harish-Chandra-Itzykson-Zuber integral. See e.g. terrytao.wordpress.com/2013/02/08/… $\endgroup$– Abdelmalek AbdesselamCommented Jan 28, 2015 at 23:04
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1 Answer
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Let $A={\rm diag}[a_1,\dots,a_n]$ and $B={\rm diag}[b_1,\dots,b_n]$. Let $\Delta(a)$ be the Vandermonde product in the $a_j$, and similarly $\Delta(b)$ be the Vandermonde product in the $b_k$. Suppose ${\rm min} \, a_j \ge {\rm max} \, b_k$. Let ${\rm d}U$ denote the normalised Haar measure on the unitary group $U(n)$. Then by Eq. (3.21) in Gross and Richards, "Total positivity, spherical series, and hypergeometric functions of matrix argument", J. Approx. Th. vol. 59 (1989): 224-246, $$ \int_U \det(A - U B U^\dagger)^p {\rm d}U = c_{n,p} {\det [ (a_j - b_k)^{p+n-1}]_{j,k=1}^n \over \Delta(a) \Delta(b)}, $$ where $c_{n,p} = \prod_{j=0}^{n-1} \binom{p+n-1}{j}^{-1}$. Setting $p=1$, subject to the condition ${\rm min} \, a_j \ge {\rm max} \, b_k$, this is the sought evaluation. See Theorem 2.3 in Kieburg, Kuijlaars and Stivigny, "Singular value statistics of matrix products with truncated unitary matrices", IMRN vol. 2016 (2016) 3392-3424 for a generalisation.
