Let O be chosen uniformly at random from the group of (N + L) × (N + L) orthogonal matrices. Denote by Õ the upper-left N × N corner of O, which we refer to as a truncation of O. In this paper we prove two conjectures of Forrester, Ipsen and Kumar (2020) on the number of real eigenvalues N (m) R of the product matrix Õ1 . . . Õm, where the matrices { Õj} m j=1 are independent copies of Õ. When L grows in proportion to N , we prove thatWe also prove the conjectured form of the limiting real eigenvalue distribution of the product matrix. Finally, we consider the opposite regime where L is fixed with respect to N , known as the regime of weak non-orthogonality. In this case each matrix in the product is very close to an orthogonal matrix. We show that E(N (m) R ) ∼ cL,m log(N ) as N → ∞ and compute the constant cL,m explicitly. These results generalise the known results in the one matrix case due to Khoruzhenko, Sommers and Życzkowski (2010).
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