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- W2018233872 abstract "n random matrix with independent Gaussian entries gij (real or complex), defined on some probability space (a, P’) and distributed according to the N(0, 1) law. In the theory of computational complexity it is of interest to consider the “random condition number” IIG;‘jl . IGnll; in particular a question about the exact order of where 11.11 denotes the operator norm on the Euclidean space and [E the expected value, was asked in (Smale, 1985) (this quantity may be inter- preted as the average “loss of precision” when solving large systems of linear equations). It is very well known that n-“*IE((G,,(J + 2 as n -+ 03; moreover, lE/G,,ll pr~~‘~) 5 C exp(-cp*n), P((lG,(j < an”*) 5 (cay2 etc. Consequently, as far as the condition numbers are concerned, esskntially the only unknown is the behavior of G;‘. Above and in what follows, C, c, etc., denote uniuersa/ (effectively computable) numerical constants, most notably independent of n; however, identical symbols may represent different numbers in different places. In this paper we deal with a more general setup, which covers, e.g., the case when 5P is endowed with the 1; - norm ~~~~~, (for some p E [ 1, ~1); we denote the corresponding operator norm by ll’llp+P (in fact our methods allow us to handle arbitrary norms, on both the domain and the range of G; see Remark 4.1). We then have the following" @default.
- W2018233872 created "2016-06-24" @default.
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- W2018233872 date "1991-06-01" @default.
- W2018233872 modified "2023-10-16" @default.
- W2018233872 title "Condition numbers of random matrices" @default.
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- W2018233872 doi "https://doi.org/10.1016/0885-064x(91)90002-f" @default.
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