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# Accession Number:

## AD0694447

# Title:

## THE ROOTS OF MATRIX PENCILS (Ay = Lambda By): EXISTENCE, CALCULATIONS, AND RELATIONS TO GAME THEORY.

# Descriptive Note:

## Research rept.,

# Corporate Author:

## CARNEGIE-MELLON UNIV PITTSBURGH PA MANAGEMENT SCIENCES RESEARCH GROUP

# Report Date:

## 1969-08-22

# Pagination or Media Count:

##
32.0

# Abstract:

## Given m x n matrices A and B, by a solution to the pencil system of A relative to B we mean a triple lambda not equal to 0, x not equal to 9, y not equal to 0, where lambda is x, x is 1 x m and y is n x 1, such that both xA-lambda B 0 and A-lambda B y 0. A canonical form is derived from which it is easy to state conditions for existence of such solutions. By further decomposing the canonical form it is shown that ordinary elimination methods and eigenvalue routines may be used to find pencil solutions. Next the relationship between matrix pencils and matrix games is studied. Finally, the relationship between game theory and dual systems of linear homogeneous equations is developed for both the real and complex cases and applied to pencil systems. Author

# Distribution Statement:

## APPROVED FOR PUBLIC RELEASE

#