Question

I have a problem bleow:x'Ax=1 where x=Cy and C is nonsingular.So my question is that what is the geometry meanning of this transform?

Answer 1

so u have
\[ \large 1=x^tAx=(Cy)^tA(Cy)=y^t(C^tAC)y \]

Answer 2

"Geomery"?

Answer 3

the expression \(x^tAx\) is a quadratic form.

do u know that?

Answer 4

I know.It also has the norm form.

Answer 5

well. quadratic forms are "conic sections" (i mean, the analogues of conic sections but in dimensions greater than 2).

whereas, since C is nonsingular (that is invertible), then the transformation \(x=Cy\) is actually an isomorphism, so the space has been spinned around the origin of reflected with respect to something or a composition of those. So the same happens to the original conic section.

Answer 6

sorry .... *or reflected

Answer 7

i can't think of anything else right now.

Answer 8

What is the diffence when C fits C'C=I.

Answer 9

When \(C^tC=I\), then the matrix C is called orthonormal.

Answer 10

I want to know what's the relationship between Euclid geometry and Affine geometry。