Examples: The following are orthogonal matrices. "How many entries of a typical orthogonal matrix can be approximated by independent normals?." Ann. We say that a square matrix U is an orthogonal matrix if the columns of U form an orthonormal basis. This says that m n= o( n/log n) is the largest order such that the entries of the first m n columns of Γ n can be approximated simultaneously by independent standard normals. We solve an open problem of Diaconis that asks what are the largest orders of p n and q n such that Z n, the p n× q n upper left block of a random matrix Γ n which is uniformly distributed on the orthogonal group O( n), can be approximated by independent standard normals? This problem is solved by two different approximation methods.įirst, we show that the variation distance between the joint distribution of entries of Z n and that of p n q n independent standard normals goes to zero provided $p_$ in probability when m n= for any α>0.
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