Relationship between anisotropic and negative/positive definite
$begingroup$
Let $K$ be an ordered field, and $(V, <->)$ a non-degen. bilinear space, where $<->$ is a bilinear form. Determine whether the following statement is true or false:
$(V,<->)$ is anisotropic <—> <-> is negative(/positive) definite.
My answer to the question is:
<—
It’s true. Since if the bilinear form is positive definite, then $<u,v> > 0 forall u,v in V$. Therefore, there’s no vector $w in V$ with $<w,w>=0$. Implying there isn’t any isotropic vector, so the space is anisotropic.
—>
(I don’t know what to do in this case, since being anisotropic doesn’t necessarily imply $<u,v> > 0 (< 0) forall u,v in V$)
I think there’s a post about it already but I don’t understand the answer there.
multilinear-algebra bilinear-form
asked Jan 8 at 19:05
M. NavarroM. Navarro
858
$endgroup$
add a comment |
$begingroup$
Let $K$ be an ordered field, and $(V, <->)$ a non-degen. bilinear space, where $<->$ is a bilinear form. Determine whether the following statement is true or false:
$(V,<->)$ is anisotropic <—> <-> is negative(/positive) definite.
My answer to the question is:
<—
It’s true. Since if the bilinear form is positive definite, then $<u,v> > 0 forall u,v in V$. Therefore, there’s no vector $w in V$ with $<w,w>=0$. Implying there isn’t any isotropic vector, so the space is anisotropic.
—>
(I don’t know what to do in this case, since being anisotropic doesn’t necessarily imply $<u,v> > 0 (< 0) forall u,v in V$)
I think there’s a post about it already but I don’t understand the answer there.
multilinear-algebra bilinear-form
asked Jan 8 at 19:05
M. NavarroM. Navarro
858
$endgroup$
add a comment |
$begingroup$
Let $K$ be an ordered field, and $(V, <->)$ a non-degen. bilinear space, where $<->$ is a bilinear form. Determine whether the following statement is true or false:
$(V,<->)$ is anisotropic <—> <-> is negative(/positive) definite.
My answer to the question is:
<—
It’s true. Since if the bilinear form is positive definite, then $<u,v> > 0 forall u,v in V$. Therefore, there’s no vector $w in V$ with $<w,w>=0$. Implying there isn’t any isotropic vector, so the space is anisotropic.
—>
(I don’t know what to do in this case, since being anisotropic doesn’t necessarily imply $<u,v> > 0 (< 0) forall u,v in V$)
I think there’s a post about it already but I don’t understand the answer there.
multilinear-algebra bilinear-form
asked Jan 8 at 19:05
M. NavarroM. Navarro
858
$endgroup$
Let $K$ be an ordered field, and $(V, <->)$ a non-degen. bilinear space, where $<->$ is a bilinear form. Determine whether the following statement is true or false:
$(V,<->)$ is anisotropic <—> <-> is negative(/positive) definite.
My answer to the question is:
<—
It’s true. Since if the bilinear form is positive definite, then $<u,v> > 0 forall u,v in V$. Therefore, there’s no vector $w in V$ with $<w,w>=0$. Implying there isn’t any isotropic vector, so the space is anisotropic.
—>
(I don’t know what to do in this case, since being anisotropic doesn’t necessarily imply $<u,v> > 0 (< 0) forall u,v in V$)
I think there’s a post about it already but I don’t understand the answer there.
multilinear-algebra bilinear-form
multilinear-algebra bilinear-form
asked Jan 8 at 19:05
M. NavarroM. Navarro
858
asked Jan 8 at 19:05
M. NavarroM. Navarro
858
asked Jan 8 at 19:05
M. NavarroM. Navarro
858
asked Jan 8 at 19:05
M. NavarroM. Navarro
858
asked Jan 8 at 19:05
M. NavarroM. Navarro
858
858
add a comment |
add a comment |
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