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On sums of three square-zero matrices

โœ Scribed by K. Takahashi

Publisher
Elsevier Science
Year
2000
Tongue
English
Weight
98 KB
Volume
306
Category
Article
ISSN
0024-3795

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โœฆ Synopsis

Wang and Wu characterized matrices which are sums of two square-zero matrices, and proved that every matrix with trace-zero is a sum of four square-zero matrices. Moreover, they gave necessary or sufficient conditions for a matrix to be a sum of three square-zero matrices. In particular, they proved that if an n ร— n matrix A is a sum of three square-zero matrices, the dim ker(A -ฮฑI ) 3n/4 for any scalar ฮฑ / = 0. Proposition 1 shows that this condition is not necessarily sufficient for the matrix A to be a sum of three square-zero matrices, and characterizes sums of three square-zero matrices among matrices with minimal polynomials of degree 2.

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## Abstract For a graph __G__ whose number of edges is divisible by __k__, let __R__(__G,Z__~k~) denote the minimum integer __r__ such that for every function __f__: __E__(__K__~r~) โ†ฆ __Z__~k~ there is a copy __G__^1^ of __G__ in __K__~r~ so that ฮฃeโˆˆ__E__(__G__^1^) __f(e)__ = 0 (in __Z__~k~). We pr