# An integral with respect to the Haar measure on a unitary group

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 $D$.)

• 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? – Vít Tuček Jan 28 '15 at 12:08
• Oops, Sorry for these typos! I edited them. Does it make sense now? Thanks. – Peter Jan 28 '15 at 17:22
• Wlog one can assume that $A$ is diagonal because the Haar measure is invariant under left- and rightmultiplication. – Johannes Hahn Jan 28 '15 at 17:28
• 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 – Yemon Choi Jan 28 '15 at 18:17
• @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/… – Abdelmalek Abdesselam Jan 28 '15 at 23:04

## 1 Answer

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.