An elementary and constructive solution to Hilbert's 17th Problem for matrices
Abstract
We give a short and elementary proof of a theorem of Procesi, Schacher and (independently) Gondard, Ribenboim that generalizes a famous result of Artin. Let
A
be an
n×n
symmetric matrix with entries in the polynomial ring
R[
x
1
,...,
x
m
]
. The result is that if
A
is postive semidefinite for all substitutions
(
x
1
,...,
x
m
)∈
R
m
, then
A
can be expressed as a sum of squares of symmetric matrices with entries in
R(
x
1
,...,
x
m
)
. Moreover, our proof is constructive and gives explicit representations modulo the scalar case.