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# Accession Number:

## ADP007105

# Title:

## Singular Values of Large Matrices Subject to Gaussian Perturbation,

# Descriptive Note:

# Corporate Author:

## AT AND T BELL LABS MURRAY HILL NJ

# Report Date:

## 1992-01-01

# Pagination or Media Count:

##
4.0

# 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 gNt. 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 EgN 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.

# Distribution Statement:

## APPROVED FOR PUBLIC RELEASE

#