Re: Inner Products
- From: Mariano Suárez-Alvarez <mariano.suarezalvarez@xxxxxxxxx>
- Date: Mon, 16 Feb 2009 04:11:00 -0800 (PST)
On Feb 16, 9:31 am, berwald.f...@xxxxxxxxxxxxxx wrote:
On 15 fev, 21:50, Mariano Suárez-Alvarez
<mariano.suarezalva...@xxxxxxxxx> wrote:
On 15 feb, 22:36, berwald.f...@xxxxxxxxxxxxxx wrote:
Hallo!
"Does the set of all inner products of a vector space have some
(algebraic, analytic, geometric) structure or interpretation?"
I saw this question in a blog... Any ideas?
By fixing a basis, you can identify a bilinear form
on a fginite dimensional space to a square matrix.
The form is non-degenerate iff the matrix has non-zero
determinant, and it is positive definite iff the determinants
of all its principal minors are positive.
This tells you that the set of inner products on a real
vector space is, in this identification, an open subset
of GL(n, R). It is this a real algebraic variety.
-- m
Thank you, Mariano.
I really liked this algebraic-geometric-topological thing!
Can you give me a reference for this, and/or some related questions?
By the way, what is the topology you are talking about?
The "usual" topology of GL(n, R) is the one it gets from
seeing it as a subset of M(n, R), the vector space of
n-by-n matrices.
-- m
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