Accession Number : ADP007105

Title :   Singular Values of Large Matrices Subject to Gaussian Perturbation,

Corporate Author : AT AND T BELL LABS MURRAY HILL NJ

Personal Author(s) : Denby, Lorraine ; Mallows, Colin

Report Date : 1992

Pagination or Media Count : 4

Abstract : Extending the work of Wachter (1978, 1980) and many others, we study the configuration of the singular values (s.v.'s) of an a by b matrix of the form X = M + sigma Z where M is a constant matrix, and the elements of Z are i.i.d., standard Gaussian, in the limit as a and b increase in constant ratio. We put N = a + b and suppose a = alpha N, b = Beta N, with (sigma of order 1 square root of N. Let the empirical distribution of the s.v.'s of X be GN, and let the corresponding moment-generating-function (m.g.f) be gN(t). These are random quantities; their distributions depend only on sigma and the empirical distribution Fn of the s.v.'s of M. We derive a differential equation that governs the evolution of E(gN) as sigma increases. In the limit as N yields infinity we can solve this equation and hence exhibit the limiting (non-random) g itself. This study was motivated by some blood-pressure data collected by a new type of transducer. It suggests a novel way of adjusting large matrices to reduce the effect of additive contamination.

Descriptors :   *MATRICES(MATHEMATICS), *ALGORITHMS, ADDITIVES, BLOOD PRESSURE, CONFIGURATIONS, CONSTANTS, CONTAMINATION, DIFFERENTIAL EQUATIONS, DISTRIBUTION, EQUATIONS, FUNCTIONS, MOMENTS, SQUARE ROOTS, STANDARDS, TRANSDUCERS, VALUE.

Subject Categories : Theoretical Mathematics

Distribution Statement : APPROVED FOR PUBLIC RELEASE