Let
)
be the set of all symmetric
matrices and
)
the set of all anti-symmetric matrices (that is matrices
with
). I claim that
^\perp=\text{AS ym}_n\left(\mathbb{R}\right))
. Indeed, assuming that you are imposing the inner product on
)
by identifying it with
you can readily prove (if you're really desperate a proof can be gleaned from
here) that
)
from where it's immediate that
\subseteq\text {Sym}_n\left(\mathbb{R}\right)^\perp)
since if
,S\in\text {Sym}_n\left(\mathbb{R}\right))
then
but with equal validity
and so
. Now, to finish the argument note that every matrix
)
may be written as
and so
=\text{Sym}_n \left(\mathbb{R}\right)+\text{ASym}_n\left(\mathbb {R}\right))
that said it's evident that
\cap\text{ASym} _n\left(\mathbb{R}\right)=\{\bold{0}\})
since if
is in the intersection then
. Thus,
= \text{Sym}_n \left(\mathbb{R}\right)\oplus\text{ASym}_n\left( \mathbb{R} \right))
and so
And so recalling that
you may conclude with a dimension argument
is
where each
has a 1 in the i,j-th position and 0's elsewhere.
. considering each of the
must be orthogonal to each and every one of the basis elements for S.
must be 0 for all i, and that for i ≠ j, the i,j-th entry of T must be the negative of the j,i-th entry.
, that is, T is anti-symmetric.
and the i,j-th entry of T,
, the j,i-th entry of T, so we have
therefore,
.
LinkBack URL
About LinkBacks
Originally Posted by qwerty1234 
