How to classify a quadratic surface?

[cookiesyum]

Homework Statement

Classify the quadratic surface:

2x^2 + 4y^2 - 5z^2 + 3xy - 2xz + 4yz = 2

Homework Equations

A quadratic form is a second degree polynomial equation in three variables which has the form

F(x,y,z) = ax^2 + by^2 + cz^2 + 2dxy + 2exz + 2fyz

where coefficients a through i are real numbers.

The curve F(x,y,z) = j can be written in the form (x^T)*A*x = j where

x = [x, y, z]

A = [a, d, e]
[d, b, f]
[e, f, c]

The Attempt at a Solution

A = [2, 3/2, -1]
[3/2, 4, 2]
[-1, 2, -5]

Then, find the eigenvectors of A-tI...check how many are positive, negative, or 0 and classify using the information? Is that the right way to proceed?

 

[yyat]

Then, find the eigenvectors of A-tI...check how many are positive, negative, or 0 and classify using the information? Is that the right way to proceed?

I think you have the right idea, but the wrong terminology: You are trying to find the eigenvalues of A by finding zeros of the polynomial det(A-tI).

 

[cookiesyum]

I think you have the right idea, but the wrong terminology: You are trying to find the eigenvalues of A by finding zeros of the polynomial det(A-tI).

Oh ok. And then, the fact that F(x, y, z) is = 2 doesn't make a difference, right? Because, it is the j that F(x, y, z) can be set equal to such that (x^T)*A*x = j?

 

[yyat]

Oh ok. And then, the fact that F(x, y, z) is = 2 doesn't make a difference, right? Because, it is the j that F(x, y, z) can be set equal to such that (x^T)*A*x = j?

The sign of the RHS does make a difference, in this case it tells you how many "sheets" the quadratic surface has (have you found the eigenvalues yet?).

 

[cookiesyum]

The sign of the RHS does make a difference, in this case it tells you how many "sheets" the quadratic surface has (have you found the eigenvalues yet?).

I get two positive and one negative eigenvalues: -5.647, 1.661, 4.986. This would be a hyperboloid of one sheet.

 

[yyat]

I get two positive and one negative eigenvalues: -5.647, 1.661, 4.986. This would be a hyperboloid of one sheet.

That's correct.

 

[cookiesyum]

That's correct.

Thanks a bunch!