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

Thompson, Gerald L.; Weil, Roman L., Jr

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

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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

Author(s):

Thompson, Gerald L. ; Weil, Roman L., Jr

Author Organization(s):

CARNEGIE-MELLON UNIV PITTSBURGH PA MANAGEMENT SCIENCES RESEARCH GROUP

Descriptive Note:

Research rept.,

Supplementary Note:

Prepared in cooperation with Chicago Univ., Ill. Committee on Information Sciences. Grant NSF-GS-1541.

Pagination:

0032

Security Markings

DOCUMENT & CONTEXTUAL SUMMARY

Distribution:

Approved For Public Release

RECORD

Collection: TR

Identifying Numbers

Report Number(s):

RR-172

Contract/Grant Number(s):

Nonr-760(24)

Project Number(s):

NR-047-048

Subject Terms

Communities of Interest:

No COI(s) Identified

Descriptor(s):

(*GAME THEORY, *MATRICES(MATHEMATICS)), DIFFERENTIAL EQUATIONS, PERMUTATIONS, ECONOMICS, THEOREMS

Field(s)/Group(s):

Theoretical Mathematics, Operations Research

Keyword(s):

EIGENVECTORS

Report Date:

1969 Aug 22