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Hi, everyone.
I am preparing for a prelim. in Diff. Geometry, and here is a question
I have not been able to figure out:
I am trying to show that Sym(n) , the set of all symmetric matrices in M_n
= all nxn matrices, is a mfld. under inclusion.
I see two possible approaches here, none of which I have been able to
get an answer from:
i) Showing Sym(n) is an open subset of M_n .
This does not seem true.
every open set containing a member ( in the topology as a subset of R^(n^2))
contains both symmetric as well as non-symmetric matrices.
ii) Figuring out directly if Sym(n) is a subspace of M_n, i.e, showing that every
open set in Sym(n) equals W/\Sym(n) , with W open in M_n ( same topology
as above).
I don't know how to do this, because I cannot figure out the open sets in Sym(n).
I don't see how to use any reasonably nice map like, e.g, the Det. to express
Sym(n) as a subset of M_n. Some matrices in Sym(n) are invertible, others are
not, so I don't see how this would work. Nor do I see how to express Sym(n)
as the inverse image under a map of constant rank.
I am out of ideas. Any suggestions?.
Thanks For Any Help.
I
I am preparing for a prelim. in Diff. Geometry, and here is a question
I have not been able to figure out:
I am trying to show that Sym(n) , the set of all symmetric matrices in M_n
= all nxn matrices, is a mfld. under inclusion.
I see two possible approaches here, none of which I have been able to
get an answer from:
i) Showing Sym(n) is an open subset of M_n .
This does not seem true.
every open set containing a member ( in the topology as a subset of R^(n^2))
contains both symmetric as well as non-symmetric matrices.
ii) Figuring out directly if Sym(n) is a subspace of M_n, i.e, showing that every
open set in Sym(n) equals W/\Sym(n) , with W open in M_n ( same topology
as above).
I don't know how to do this, because I cannot figure out the open sets in Sym(n).
I don't see how to use any reasonably nice map like, e.g, the Det. to express
Sym(n) as a subset of M_n. Some matrices in Sym(n) are invertible, others are
not, so I don't see how this would work. Nor do I see how to express Sym(n)
as the inverse image under a map of constant rank.
I am out of ideas. Any suggestions?.
Thanks For Any Help.
I