# Singular Value Statistics of Matrix Products with Truncated Unitary Matrices

Kieburg M, Kuijlaars ABJ, Stivigny D (2016) *International Mathematics Research Notes* (11): 3392-3424.

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We prove that the squared singular values of a fixed matrix multiplied with a truncation of a Haar distributed unitary matrix are distributed by a polynomial ensemble. This result is applied to a multiplication of a truncated unitary matrix with a random matrix. We show that the structure of polynomial ensembles and of certain Pfaffian ensembles is preserved. Furthermore we derive the joint singular value density of a product of truncated unitary matrices and its corresponding correlation kernel which can be written as a double contour integral. This leads to hard edge scaling limits that also include new finite rank perturbations of the Meijer G-kernels found for products of complex Ginibre random matrices.

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Kieburg M, Kuijlaars ABJ, Stivigny D. Singular Value Statistics of Matrix Products with Truncated Unitary Matrices.

*International Mathematics Research Notes*. 2016;(11):3392-3424.Kieburg, M., Kuijlaars, A. B. J., & Stivigny, D. (2016). Singular Value Statistics of Matrix Products with Truncated Unitary Matrices.

*International Mathematics Research Notes*(11), 3392-3424. doi:10.1093/imrn/rnv242Kieburg, M., Kuijlaars, A. B. J., and Stivigny, D. (2016). Singular Value Statistics of Matrix Products with Truncated Unitary Matrices.

*International Mathematics Research Notes*, 3392-3424.Kieburg, M., Kuijlaars, A.B.J., & Stivigny, D., 2016. Singular Value Statistics of Matrix Products with Truncated Unitary Matrices.

*International Mathematics Research Notes*, (11), p 3392-3424. M. Kieburg, A.B.J. Kuijlaars, and D. Stivigny, “Singular Value Statistics of Matrix Products with Truncated Unitary Matrices”,

*International Mathematics Research Notes*, 2016, pp. 3392-3424. Kieburg, M., Kuijlaars, A.B.J., Stivigny, D.: Singular Value Statistics of Matrix Products with Truncated Unitary Matrices. International Mathematics Research Notes. 3392-3424 (2016).

Kieburg, Mario, Kuijlaars, Arno B. J., and Stivigny, Dries. “Singular Value Statistics of Matrix Products with Truncated Unitary Matrices”.

*International Mathematics Research Notes*11 (2016): 3392-3424.
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