JP Journal of Algebra, Number Theory and Applications
Volume 3, Issue 3, Pages 435 - 445
(December 2003)
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ADDITIVE
{1,
2}-INVERSE
PRESERVERS ON spaces of SYMMETRIC MATRIces OVER
FIELDS OF CHARACTERISTIC NOT 2
Xian Zhang (P. R. China) and Chong-Guang Cao (U. K.)
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Abstract: Suppose F is a field of
characteristic not 2. Let Mn(F)
and Sn(F)
be the n´n
full matrix space and symmetric matrix space
over F, respectively. This paper
investigates that (i) a nonzero additive map f
from Sn(F)
to Mn(F)
preserves -inverses
of matrices if and only if there exist a number eÎ{–
1, 1}, a nonsingular matrix PÎMn(F)
and an injective field endomorphism d on F such
that f(X)=ePXdP–1
for any X
Î
Sn(F); and (ii) a nonzero additive map f from
Sn(F)
to itself preserves {1,
2}-inverses
of matrices if and only if f has the form
in (i) except that P satisfies PTP
= zIn for some z
Î
F. |
Keywords and phrases: field, characteristic, -inverse,
additive preserver, full matrix space,
symmetric matrix space. |
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